A differential extension of Descartes' foundational approach: a new balance between symbolic and analog computation
AA differential extension of Descartes’ foundationalapproach: a new balance between symbolic andanalog computation
Pietro Milici Universit´e de Bretagne Occidentale, Brest, France; [email protected]
Abstract In La G´eom´etrie , Descartes proposed a “balance” between geometricconstructions and symbolic manipulation with the introduction of suitableideal machines. In modern terms, that is a balance between analog andsymbolic computation.Descartes’ geometric foundational approach (analysis without infini-tary objects and synthesis with diagrammatic constructions) has beenextended beyond the limits of algebraic polynomials in two different pe-riods: by late 17th century tractional motion and by early 20th century differential algebra . This paper proves that, adopting these extensions,it is possible to define a new convergence of machines (analog computa-tion), algebra (symbolic manipulations) and a well determined class ofmathematical objects that gives scope for a constructive foundation of(a part of) infinitesimal calculus without the conceptual need of infinity.To establish this balance, a clear definition of the constructive limits oftractional motion is provided by a differential universality theorem . In La G´eom´etrie , Descartes proposed a “balance” between geometric construc-tions and symbolic manipulation with the introduction of suitable ideal ma-chines. In particular, Cartesian tools were polynomial algebra (analysis) and aclass of diagrammatic constructions (synthesis). This setting provided a classi-fication of curves, according to which only the algebraic ones were considered“purely geometrical.” Thanks to this approach, geometrical intuition was nolonger necessary in proving new properties, because the “method” (suitable al-gorithms in the analysis) permitted to lead the thought: in modern term, therewas the seed of automated reasoning. Descartes’ limit was overcome with a gen-eral method by Newton and Leibniz introducing the infinity in the analyticalpart, whereas the synthetic perspective gradually lost importance with respectto the analytical one—geometry became a mean of visualization, no longer ofconstruction.Descartes’ foundational approach (analysis without infinitary objects andsynthesis with diagrammatic constructions) has, however, been extended be-1 a r X i v : . [ m a t h . HO ] S e p ond algebraic limits, albeit in two different periods. In the late 17th century,the synthetic aspect was extended by tractional motion (construction of tran-scendental curves with idealized machines). In the first half of the 20th century,the analytical part was extended by differential algebra , now a branch of com-puter algebra in which, informally speaking, the indeterminates are not numbersbut continuous (and differentiable) functions. This paper seeks to prove thatit is possible to obtain a new balance between these synthetic and analyticalextensions of Cartesian tools for a class of transcendental problems.A reason for a renewing of the Cartesian program concerns the historicalevolution of mathematical objects. Mathematics can be considered as basedon two cornerstones: arithmetics (symbolic manipulation of discrete elements)and geometry (constructions based on idealized continuous behaviors). Thesynthetic components of such approaches are respectively digital and analogcomputation, and the mutual relationship between these kind of constructionsprovides a cognitive richness that leads the main steps of mathematical evolu-tion. When arithmetic and geometric strengths are unbalanced, their mutualconversions can constitute a challenge to overreach their own limits, as evincedfrom a linguistic perspective in [Kvasz, 2008, Ch. 1] (even though withoutdistinguishing between the constructive and mere visual power of geometry).Being today mainstream mathematics too much oriented toward arithmetics,our aim is to resume the commitment toward geometric constructions. We aredealing with geometry instead of general analog computing because, accordingto Descartes’ perspective, the primitive bases of our knowledge have to be intu-itively clear, and the simple components of geometric ideal machines minimizethe physical complexity and the cognitive requirements. From this perspective,the millennial endurance of Euclid’s geometric paradigm can be justified by thewide application of its synthetic solutions, by the rigor of its analysis, but alsoby the concreteness of its constructive tools (segments and circles can be tracedby ruler and compass).Concerning the organization of this work, we show a new convergence ofmachines (analog computation, section 2), algebra (symbolic manipulations,section 3), and geometry (constructed mathematical objects, section 4) that,together with a problem solving method (section 5), gives scope for a foundationof (a part of) infinitesimal calculus without the conceptual need of infinity . Toestablish this balance, a clear (historically missing) definition of the constructivelimits of tractional motion is provided by a differential universality theorem .The peculiarity of this work lies in the attention to the constructive role ofgeometry as idealization of machines for foundational purposes. This approach,after the de-geometrization of mathematics, is far removed from the mainstream An objection to such avoidance of infinity could be that we cannot really avoid the in-finite in the analytic part, because to define continuous functions at the basis of differentialalgebra we need limits or similar tools. With regard to this objection, we claim that, evenif one considers continuity expressible only through infinitary tools, the allowed operationsin differential algebra remain in the field of a finitist symbolic manipulation (in fact, differ-ential algebra is nowadays considered a field of computer algebra). The constructive role ofinfinity in differential algebra is avoidable as it is in the analysis of polynomial algebra. Inclassical algebra, indeterminates assume values on the field of the real numbers, the definitionof which requires infinity, but algebra remains finite because it does not deal with generalreal numbers, one simply makes manipulations and can control only a countable subset ofreal numbers. Similarly, differential algebra does not deal with the definition of continuousfunctions: the underlying requirement is to manipulate symbols that represent such functions. igure 1: [Left] The heavy body is B , with initial position B , the string is a , and the otherend of the string is A , with initial position A . Moving A along r , B describes the tractrix(obviously, the movement is not reversible because of the non-rigidity of the string). Notehow a is tangent to the curve at every point.[Right] Reconstruction of Perks’ instrument for the tractrix [Pedersen, 1963, p. 17], copyrightlicense number 4518140353283. One can see the wheel taking the place of the load and a barinstead of the string. discussions of mathematics, especially regarding foundations. However, thoughforgotten these days, the problem of defining appropriate canons of constructionwas very important in the early modern era, and heavily influenced the definitionof mathematical objects and methods. According to Bos’ definition in Bos[2001], these are exactness problems for geometry. The problem of extending geometry beyond Cartesian limits was dominant be-tween 1650 and 1750 Bos [1988], and in this section, we shortly deal with it.If direct tangent problems are present since the classical period, it was onlyin the second half of the 17th century that the inverse ones appeared. Themain difference between direct and inverse tangent problems is the role of thecurve: in the direct case it is given a priori , while in the second the curveis sought given some properties that its tangent has to satisfy. Even thoughbeyond Cartesian geometry, to legitimate solutions of inverse tangent problemsthere was the introduction of certain machines, intended as both theoretical andpractical instruments, able to trace such curves. The first documented curvesconstructed under tangent conditions were physically realized by the tractionof a string tied to a load, which is why the study of these machines was named tractional motion . Consider the following example:On a horizontal plane, a small heavy body (subjected to the frictionon the plane) is tied with an ideally weightless non-elastic string, andimagine (slowly) pulling the other end of the string along a straightline drawn on the plane.Because of the friction on the plane, the body offers resistance to the pulling ofthe string: if the motion is slow enough to neglect inertia, the curve describedby the body is called a tractrix . Examining the left of Fig. 1, we can see howthe curve is traced thanks to the property that the string is constantly tangentto the curve. 3uring this period, mathematicians like Huygens began to consider instru-ments that, like the handlebars of a bike, could guide the tangent of a curve(in analytical mechanics terms, they introduced non-holonomic constraints), inorder to avoid inessential physical complications and so to consider tractionalmotion as “pure geometric.” Tractional motion suggested the possibility ofconstructing curves by imposing tangential conditions, generalizing (in a non-Cartesian way) the idea of geometrical objects, and constructing with new toolsnot only algebraic curves, but also some transcendental ones (seen as solutionsof differential equations). During this period, the development of geometricalideas often corresponded to the practical construction (or at least conception)of mechanical machines able to embody the theoretical properties, and thus ableto trace the curves. Concerning practical machines, we recall those introducedin Perks [1706, 1714] (for the machine for the tractrix see the right of Fig. 1),which, for the first time, included a “rolling wheel” to guide the tangent. Amore influential role for similar machines was played by the ones proposed in[Poleni, 1729,
Ad Iacobum Hermannum Epistola ]. An overview of such machinesis visible in Crippa and Milici [2019]. As deepened below, the wheel is able tosolve the inverse tangent problem because it avoids the lateral motion of itscontact point. Furthermore, wheels imply less physical problems than draggingloads (e.g. inertia).While questions about exactness in geometric constructions were so impor-tant in the early modern period, they disappeared in the 18th century becauseof the general affirmation of symbolic procedures, later considered autonomousfrom geometry. But, in contrast to what happened for algebraic curves, trac-tional motion did not reach a widely affirmed canon of constructions. Moreover,due to the change in paradigm, the geometric-mechanical ideas behind trac-tional machines remained forgotten for centuries, even for practical purposes,and were independently re-invented in the late 19th century, when they wereused to build some grapho-mechanical instruments of integration (integraphs) toanalogically compute symbolically non-solvable problems (for further reading,see Blasjo [2017], Tourn`es [2009]).
Leaving history behind, the goal of this part is to clearly define the componentsthat can be used to obtain devices that implement certain tangent propertieson a plane. Such clarification about the machines to be accepted in tractionalconstructions was historically missing: these ideal devices have only recentlybeen defined with the so called tractional motion machines (or TMMs) Milici[2012a, 2015].We define the mechanical components that are allowed in a modern inter-pretation of tractional motion: we adopt these because they seem to give agood compromise between the simplicity of the components (two instrumentsand two constraints) and that of the assembled machines (even if the proposedcomponents are not minimal [Milici, 2012a, Section 2]). Machines obtained as-sembling these components (to be considered on a plane that can be infinitelyextended) can be considered as an extension of Kempe’s linkages: Kempe [1876]stated the so-called Kempe’s universality theorem , that every bounded portionof a planar algebraic curve can be traced by linkages made of jointed finite-length rods (the proof was flawed, a corrected proof is in Kapovich and Millson4 igure 2: A wheel rolling while following any regular curve has the property that its own direction (represented in the picture by a bar) is always tangent to the curve. [2002]). In section 5.2 we provide a generalization of Kempe’s result for TMMswith a differential universality theorem . Now, let us define the components of
Tractional Motion Machines . • We adopt rods , and assume these have perfect straightness and negligiblewidth. They can be finite or infinitely extensible: in both cases they aredifferent from the Euclidean segments and straight lines, because they arenot statically traced objects but planar rigid bodies (mechanical entitieswith three degrees of freedom, two characterizing the position of a specificpoint and the third identifying the slope with respect to a fixed line). • It is possible to put some carts on a rod, each one using the rod as a rail:a cart has one degree of freedom once placed on a rod (the cart can onlymove up and down the rod). • The joint is a constraint between fixed points of two (or more) differentobjects (here, “object” refers to the plane, a rod, or a cart). Once the jointhas been applied, jointed objects can only rotate around their commonpoint (note that, in general, the junction point does not have to be fixedon the plane). • Finally, we have the non-holonomic constraint, the wheel : once a rod r and a point S on r have been selected, we can set a wheel at S thatprevents S itself moving perpendicularly to r (considering the motion of S with respect to the plane). Technically, this is as if we put a fixedcaster (oriented like r ) at S , with its wheel rotating without slipping onthe plane. As evinced since the construction of the tractrix, the avoidanceof lateral motion in the rod at a point is strongly related to the tangent. Ifwe consider the caster wheel as a disk rolling perpendicularly to the baseplane, the projection of the disk surface is always tangent to the curvedescribed by the disk contact point (see Fig. 2). Thus, the rod is tangentto the curve traced by the wheeled point, having the same direction as thecaster wheel.Like Kempe’s linkages, even TMM tools are assumed to be ideal (we do notcare about physical inaccuracies), and we do not consider problems related tocollisions of rods or of different carts on the same rod. Once specified suchdetails, these components can be used to assemble machines whose motion on5 igure 3: Schematic representation of the components: there are two rods ( r and s ) joined at Q . On r , there is also a cart P (the arrows stand for the possible motions the cart can have)and a wheel S (the gray thick line ideally represents the projection of a wheel). a plane is purely kinematic (just kinematic constraints, with no attention toother physical interrelations). For a diagrammatic representation of assembledcomponents, see Fig. 3.As an important remark note that, differently from the general setting oflinkages, we are not only looking for a mechanical method to define geometricalobjects, but for a computational model not involving infinity from a foundationalperspective. That means that we cannot accept any general distance betweenpoints fixed on a rod, because that would imply the introduction of real numbersand so of uncountable sets. As deepened in the following section, a simplesolution is to introduce an arbitrary unit length and, for any point P fixed on arod r , to admit the constructability of the points on r distant one unit from P .This unit-distance primitive operation is necessary because in our model thereis no compass available to transfer lengths. Once introduced the components, we have to define how to properly assemblethem with an adequate language (as an exemplary formalization of geometricconstructions in a computational language see Huckenbeck [1989]). In our lan-guage, points are enumerated by natural numbers ( P , P , . . . ), and the index ofthe point P n is n . These points, if not differently imposed by other constraints,can freely move on the plane. However, to construct a TMM, we have to startfrom a certain number of given points that are fixed on the plane. For a min-imal definition, we consider as given two distinct fixed points P and P : weintroduce the unit as their distance.Rods are represented by couples of natural numbers: by r ( i, j ) we mean thata rod is introduced in the point P i . However, we can consider many rods jointedin the same point, hence the number j distinguishes between them. E.g. let r be the first rod in our construction to be jointed in P , s the second rod jointedin P : in our language r is represented by r (3 ,
0) and s by r (3 , r ( i, j ) theordered values ( i, j ) are named the indices of the rod.There are three admissible instructions: • onRod(k,i,j) with k, i, j ∈ N . It imposes that the point P j lies on therod r ( k, i ). Recalling the components, it implies to put a cart on a rod6 P j can move along r ( k, i )). • dist(k,i,n,j) with k, i, j ∈ N , n ∈ Z . It imposes that the point P j lieson the rod r ( k, i ) at a distance n from P k . We have to note that n can alsobe a negative integer, thus we need to define an orientation of the rod, asdeepened below. This instruction implies the introduction of a joint. • wheel(k,i) with k, i ∈ N . It imposes a wheel on the point P k oriented asthe rod r ( k, i ).We have to spend few more words about the introduction of integer distanceson rods by dist . The starting point is that, once introduced the unit, we canconstrain any two points on a rod to be at the distance of a unit. Hence, wecould consider the instruction unitDist(k,i,j) with k, i, j ∈ N that imposes P j to stay on r ( k, i ) at a distance of one unit from P k . Without the moregeneral dist , we could obtain the point P l on r ( k, i ) at a distance of 2 unitsfrom P k by introducing a new rod r ( j,
0) with P k lying on it, and then imposing unitDist(j,0,l) . In this way, the point P l not coincident with P k is thesought point. Iterating this construction, we can consider any integer distanceon a rod; however, that would require to impose conditions on coincidence ofdifferent points (e.g. P l (cid:54)≡ P k ), thus we prefer to adopt dist with any integerdistance and an orientation.Furthermore, about the possibility of introducing an orientation, we have toprecise that the specific orientation of a rod is not important, it is importantthat the orientation of the rod is coherent with all its points: e.g. given theinstructions dist(0,0,2,2) and dist(0,0,-1,3) , the distance between P and P has to be 3, not 1. As an example to practice with dist , consider in Fig. 4the visual representation of the following code (commented on the right). Notethat, using only dist , we get exactly Kempe’s linkages with integer-length rods. Example 1.Code.
Considering the two given fixed points P , P , let us introduce the points P , P at unary distance to, respectively, P and P . Set the distance between P and P equal to 2 units, and consider their middle point P . dist(0,0,1,2) consider P on the rod r (0 , at unary distance from P dist(1,0,1,3) consider P on the rod r (1 , at unary distance from P dist(2,0,2,3) set the distance between P and P equal to 2 units along r (2 , dist(2,0,1,4) P is the point on r (2 , distant 1 unit from P Note that, even though there are two point on r (2 ,
0) distant 1 unit from P , P is uniquely defined because it has to be in the ray −−−→ P P ( P has distance+2 from P , thus with the same sign of the distance of P ). The other pointone unit away from P is P defined by dist(2,0,-1,5) . About indices, notethat two points with different indices can coincide: adding dist(3,0,2,2) and dist(3,0,1,6) would imply that the point P concides with P in every con-figuration.Once defined the instructions, we can consider their analytical conversion.That permits us to use computer algebra as a tool of automated reasoning on thebehavior of our machines. First of all, when introducing a rod r ( k, i ), besides P k we have to consider an auxiliary point Q k,i to determine the orientation of the7 igure 4: A simple machine without carts and wheels. The dotted line represents the locusdefined by the point P . See footnote 5 at page 19 for an analytical study. rod. This point has to satisfy only the property of being one unit away from P k (informally, Q k,i is the point P j s.t. dist(k,i,1,j) ). Analytically, consideringa system of Cartesian coordinates s.t. P = (0 ,
0) and P = (1 , x ( P ) , y ( P ) respectively for the abscissa and the ordinate of thepoint P , we can introduce Q k,i by the equation:( x ( P k ) − x ( Q k,i )) + ( y ( P k ) − x ( Q k,i )) = 1 . With Q k,i , it is easy to convert instructions in equations: onRod(k,i,j) be-comes( x ( P k ) − x ( Q k,i ))( y ( P j ) − y ( Q k,i )) = ( x ( P j ) − x ( Q k,i ))( y ( P k ) − y ( Q k,i ))and dist(k,i,n,j) is expressed by (cid:40) x ( P j ) = x ( P k ) + n ( x ( Q k,i ) − x ( P k )) y ( P j ) = y ( P k ) + n ( y ( Q k,i ) − y ( P k )) . For the wheel constraint, we have to consider a point not only as a coupleof coordinates, but as a couple of functions. As typical in physics, consider P k = ( x k ( t ) , y k ( t )), i.e. consider the Cartesian coordinates of the point infunction of the time. The instruction wheel(k,i) poses the condition that P k cannot move perpendicularly to r ( k, i ): so, considering P (cid:48) k = (cid:0) ddt x k , ddt y k (cid:1) , P (cid:48) k has to be parallel to Q k,i − P k . Thus, omitting the dependence on t andconsidering P (cid:48) k = ( x (cid:48) ( P k ) , y (cid:48) ( P k )), the wheel constrain becomes y (cid:48) ( P k )( x ( Q k,i ) − x ( P k )) = x (cid:48) ( P k )( y ( Q k,i ) − y ( P k )) . Thus, both wheel constraints and the other instructions are translatable inpolynomials in the variables and their derivatives: as deepened in the algebraicpart (section 3), such polynomials are named differential polynomials and con-stitute the basis for differential algebra.In the following section we introduce subroutines to solve some problemsabout constructions. When using subroutines, we also need to define the in-structions return(i) and newPoint(i) (with i ∈ N ). Note that, once fixed theinput, each implementation can be obtained without subroutines and the lasttwo instructions (they are not adding any primitive to the model). However,subroutines are useful to give general constructions in function of inputs.About return(a) , it returns the natural number a to the program callingthe subroutine. 8he other instruction, newPoint(i) , returns a natural number not previ-ously used to enumerate an already introduced point (i.e. first and third argu-ments of onRod , first and forth arguments of dist , first argument of wheel ).To stay compact, the shorter notation [i] stands for newPoint(i) . This in-struction comes in useful because, in a subroutine, we don’t know which naturalnumbers are yet free to introduce new points. Let n be the highest value usedas index of a point before the execution of the instruction: newPoint returnsthe integer n + 1. But later we may need to recall the newly introduced index,and we can use multiple newPoint : that’s why it was necessary to add the index i (e.g. onRod([0],0,[1]) constrains the point P [1] to be on the rod r ([0] , P i without knowing howmany rods have already been joined in it, we can consider a new point P [ k ] (as-suming that previously we have introduced P [0] , P [1] , . . . , P [ k − ) coincident with P i . The new rod r ([ k ] ,
0) can be defined by the instruction dist([k],0,0,i) .To conclude, we have to remark that the use of newPoint can create some un-wanted situations. For example, in a subroutine, it can happens that newPoint creates an index that is used in the following instructions as a fixed index ofa point. To avoid that, it might be useful to have a set of indices reserved forthese local construction points, but we leave such possible optimization (as areal software implementation) to future works.
It is also interesting to consider TMMs without wheels. In this case everyconstraint can be translated in algebraic polynomial equations, so we call theobtained machines “algebraic.” There are algebraic machines that cannot beconsidered as Kempe’s linkages, because rods are not only used to constrain afixed length between two junction points, but also to allow a point to move alonga straight line (thanks to carts). With only Kempe’s linkages to trace a straightline is not trivial at all, the problem was solved in an approximated way byWatt, and later exactly solved by Peaucellier (see, for example, [Demaine andO’Rourke, 2007, pp. 29-30]). Algebraic machines can be somehow consideredas more adherent to Descartes’ machines for geometry than to Kempe’s linkages(Descartes used machines in which straight components were allowed to slidealong other straight components). Furthermore, the introduction of “extensi-ble” rods, allows to trace not only finite part of algebraic curves, but wholecontinuous branches of algebraic curves (as already observed, Kempe’s linkagescan construct only bounded portions of algebraic curves).Before solving some problems to perform sum and product with algebraicmachines, we need to remark that the components of algebraic machines arenot part of Euclid’s geometry: they are movable mechanical parts, not statictraces on a plane, so constructions have to work dynamically. Thus, even thoughproblems like the following ones can be easily solved with ruler and compass if wesubstitute “rod” with “segment,” we need new constructions for our mechanicalsetting. All the constructions dynamically work in every configuration of theinputs, and in general input objects are points and rods that can move.
Problem 1 (perpendicular) . Given a rod r and a point P , construct a rod s perpendicular to r passing through P . Code.
Let the rod r and P be in our language respectively r ( k, i ) and P j . The igure 5: Construction of the rod r ([0] ,
0) perpendicular to r ( k, i ) passing through P j . subroutine returns the number a such that r ( a, is the sought rod. Note thatwe can guarantee that in the rod r ( a, the second index is 0 because the point a is newly introduced by the subroutine. perp(k,i,j) signature of the subroutine onRod(k,i,[0]) consider a new point P [0] on r ( k, i ) dist([0],0,4,[1]) consider a new point P [1] on r ([0] , dis-tant 4 units from P [0] dist([0],1,3,[2]) consider a new point P [2] distant 3 unitsfrom P [0] dist([2],0,5,[1]) constrain P [1] to stay 5 units away from P [2] onRod([0],0,j) constrain P j to stay on r ([0] , onRod(k,i,[2]) constrain P [2] to stay on r ( k, i ) return([0]) the sought rod is r ([0] , , return [0] Proof.
Construct a right triangle by the junction of a Pythagorean triple as rodlengths (e.g. connect three rods respectively of length 3-4-5). Consider an in-finitely extensible rod s for one of the catheti, and make the other cathetus slideon r with two carts on the vertices. Pose another cart on s in correspondenceof P . As visible in Fig. 5, that solves the problem. Problem 2 (parallel) . Given a rod r and a point P , construct a rod parallel to r passing through P . Code.
Let the rod r and P be respectively r ( k, i ) and P j . The subroutine returnsthe number b such that r ( b, is the sought rod. parall(k,i,j) signature of the subroutine return(perp(perp(k,i,j),0,j)) let’s start from the inner subroutines: perp ( k, i, j ) returns the index a s.t. r ( a, is a rod per-pendicular to r ( k, i ) passing through P j . Then, perp ( a, , j ) returns the index b s.t. r ( b, is arod perpendicular to r ( a, passing through P j .Such b is returned.Proof. According to Problem 1 we can construct a rod s perpendicular to r passing through P , and similarly we can construct a rod t perpendicular to s passing through P : t solves the problem.10 igure 6: Schema of the construction of the point ( y, x ) given the point ( x, y ) using rodsparallel and perpendicular to the x-axis and to r (2 ,
0) (rod through (1 ,
0) and (0 , In the following problems we adopt Cartesian coordinates to simplify the no-tation. We have already introduced the points P and P respectively of coordi-nates (0 ,
0) and (1 , r (0 ,
0) as the usually oriented x-axis(by dist(0,0,1,1) ), and the point P at unary distance from P on the rodperpendicular to r (0 ,
0) passing through P (by dist(perp(0,0,0),0,1,2) ).Considering the x-axis oriented horizontally to the right, the possible positionsof P can be above or below the x-axis. However, our constructions works forboth the possibilities, it only changes the orientation of the coordinate system.In the following figures we always consider the standard positive orientation(while the positive x-axis points right, the positive y-axis points up), but all theresults remain with the negative orientation. Problem 3 (inverse) . Given a point of Cartesian coordinates ( x, y ) , constructa point of coordinates ( y, x ) . Code.
Assuming that P , P , P are respectively (0 , , (1 , , (0 , , let P i =( x, y ) be the input point. The subroutine returns the index of the point ( y, x ) . inv(i) signature of the subroutine onRod(2,0,1) the rod r (2 , passes through P onRod(perp(0,0,i),0,[0]) P [0] is on the rod through P i parallel to they-axis onRod(perp(2,0,0),0,[0]) P [0] also is constrained to lie on the bisectorof the first and third quadrant onRod(parall(0,0,[0]),0,[1]) P [1] is on the rod through P [0] parallel to thex-axis onRod(parall(2,0,i),0,[1]) P [1] is also constrained to lie on the rod parallelto r (2 , through P i return([1]) the sought point is P [1] Proof.
As visible in Fig. 6, starting from the point ( x, y ) and calling r (2 , ,
0) and (0 , x, x ) andfinally ( y, x ) by intersecting rods parallel and perpendicular to the x-axis andto r (2 , x , y ), we can constructthe points of coordinates ( x ,
0) and ( y , x ,
0) and ( y , x , y ). Thus, to11 igure 7: Construction of the point ( x + x ,
0) given the points ( x , y ) and ( x , y ). represent variables in algebraic machines, we can interpret them simply as pointsmoving on the abscissa. Algebraically, such variables are real values. It’s timeto show how to perform the internal binary operations of sum, difference andmultiplication for abscissas of points. Problem 4 (sum) . Given two points of Cartesian coordinates ( x , y ) and ( x , y ) , construct a point of coordinates ( x + x , . Code.
Consider the x-axis r (0 , , and let ( x , y ) , ( x , y ) respectively be P i , P j .The subroutine returns the index of the point ( x + x , . sum(i,j) signature of the subroutine onRod(parall(0,0,inv(i)),0,[0]) a new point P [0] lies on y = x onRod(perp(0,0,j),0,[0]) P [0] has coordinates ( x , x ) onRod(inv(i),0,i) a new rod r ( inv ( i ) , passes through ( x , y ) and ( y , x ) onRod(parall(inv(i),0,[0]),0,[1]) the new point P [1] lies on the rod parallelto r ( inv ( i ) , passing through P [0] onRod(0,0,[1]) the point P [1] is constrained to lie on thex-axis return([1]) the sought point P [1] has coordinates ( x + x , Proof.
According to Problem 3, consider the point of coordinates ( y , x ). Thanksto Problem 2 we can consider the rod r parallel to the x-axis passing through( y , x ) and the rod s parallel to the y-axis passing through ( x , y ). By carts wecan identify the point ( x , x ) (the one lying on both r and s ). Finally we canconsider the rod t joined in ( x , x ) parallel to the rod passing through ( x , y )and ( y , x ): the point in the intersection of t and the x-axis has coordinates( x + x ,
0) (cf. Fig. 7).
Problem 5 (difference) . Given two points of coordinates ( x , y ) and ( x , y ) ,construct a point of coordinates ( x − x , . Code.
Consider the x-axis r (0 , , and let ( x , y ) , ( x , y ) respectively be P i , P j .The subroutine return the index of the point ( x − x , . igure 8: Construction for the multiplication of the abscissae of ( x , y ) and ( x , y ). Toobtain ( x · x ,
0) we can invert the coordinates of P [2] = (0 , x · x ) . diff(i,j) signature of the subroutine onRod(0,0,[0]) P [0] lies on the x-axis onRod(0,0,[1]) P [1] lies on the x-axis onRod(perp(0,0,i),0,[1]) P [1] has coordinates ( x , dist(sum([0],j),0,0,[1]) the abscissa of P [1] has to be the sum of theabscissae of P [0] and P j return([0]) the sought point P [0] has coordinates ( x − x , Proof.
Consider P [0] , P [1] on the x-axis, constrain P [1] to have the same abscissaof P j (thus P [1] = ( x , P [1] to have asabscissa the sum of the abscissae of P [0] and P i : hence P [0] = ( x − x , Problem 6 (multiplication) . Given two points of Cartesian coordinates ( x , y ) and ( x , y ) , construct a point of coordinates ( x · x , . Code.
Let (0 , , (1 , , ( x , y ) , ( x , y ) respectively be P , P , P i , P j , and let thex-axis be r (0 , . The subroutine return the index of the point ( x · x , . mult(i,j) signature of the subroutine onRod(parall(0,0,inv(j)),0,[0]) a new point P [0] lies on the rod parallel to thex-axis passing through ( y , x ) onRod(perp(0,0,1),0,[0]) P [0] is constrained to have as abscissa onRod([0],0,0) the origin lies on the rod r ([0] , onRod(perp(0,0,i),0,[1]) a new point P [1] lies on a rod perpendicular tothe x-axis passing through P i onRod([0],0,[1]) the point P [1] is constrained to lie on the rodpassing through the origin and (1 , x ) onRod(parall(0,0,[1]),0,[2]) P [2] is constrained on the rod parallel to thex-axis passing through ( x , x · x ) onRod(perp(0,0,0),0,[2]) P [2] has coordinates (0 , x · x ) return(inv([2])) the sought point is the one obtained invertingabscissa and ordinate of P [2] Proof.
As it happens in Descartes’ interpretation of multiplication (the length c = a · b is given by the proportion a : c = 1 : b ), we need to use the unit13ength. According to Problem 3, construct the point ( y , x ). Considering theintersection of the rod parallel to the x-axis through ( y , x ) and the rod parallelto the y-axis through (1 , , x ). We canintroduce the rod r joined in (1 , x ) passing through the origin (0 , r with the rod parallel to the y-axis passing through( x , y ) determines the point ( x , x · x ). If we project it on the y-axis we obtain(0 , x · x ), that for Problem 3 gives us the wanted ( x · x , In Cartesian geometry, polynomial algebra is used as finite tool for analysis.In the proposed differential extension, we substitute polynomials with differen-tial polynomials, machines for algebraic constructions (algebraic machines) withTMMs, and algebraic curves with manifolds of zeros of differential polynomials.In this section, we delve deeper into the analytical counterpart of TMMs, the differential algebra , specifically differential elimination . The peculiarity of thisapproach is that it is algorithmically implementable (it is part of computer alge-bra): its finite symbolic manipulation does not need any reference to infinitaryobjects (as it happens in infinitesimal calculus). These algebraic tools allowanswering some questions about TMMs in section 5.Differential algebra started with Ritt [1932], where Ritt introduced suitablealgebraic tools for differential equations. These results have been reformulatedin more and more algebraic way in Ritt [1950] and later in Kolchin [1973]. Underboth Ritt and Kolchin, basic differential algebra was developed from a construc-tive view point and the foundation they built has been advanced and extendedto become applicable in symbolic computation, mainly thanks to the passagefrom old constructive methods (Ritt-Seidenberg algorithm of Seidenberg [1956])to more recent computational complexity optimizations with Gr¨obner bases-likeapproach (firstly introduced in Carr`a Ferro [1989]) . For a brief introduction tothese computational problems and the relative historical evolution see [Boulier,2007, pp. 110–111].The aim of differential algebra is to provide an algebraic theory for differen-tial equations both ordinary or with partial derivatives. In particular, its toolsand notations are an extension of commutative algebra. To give a short intro-duction to differential algebra, we recall [Boulier, 2007, pp. 112–116] because of Such ideas have been developed in Computer Algebra Systems, e.g. in the package
DifferentialAlgebra (see ). This package is based on the software
BLAD (standing for
Bib-lioth`eques Lilloises d’Alg`ebre Diff´erentielle ), developed in the C programming language byF. Boulier. While DifferentialAlgebra is a package for the commercial software
Maple , the
BLAD software is freely available online at . Furthermore, even though yet under construction, a differential-algebraexperimental package for the open software
SageMath can be freely downloaded at the link https://trac.sagemath.org/ticket/13268 . Another free alternative is
ApCoCoA , availableat apcocoa.org (for our purposes, we have to cite the package diffalg ), a software packagebased on
CoCoA , http://cocoa.dima.unige.it . differential polynomials .In this case, out of the binary operations of sum and multiplication, we haveto introduce the unitary operation of derivation . The derivation must be dis-tributive over addition (for every a, b ∈ R it holds D ( a + b ) = D ( a ) + D ( b )) andmust obey the product rule (also called Leibniz rule , D ( ab ) = D ( a ) b + aD ( b )).Adopting the standard notation for ordinary derivatives, from now on we write a (cid:48) instead of D ( a ).For our purposes, the coefficients of such differential polynomials are ratio-nal numbers. Specifically, given a finite set U of variables, named differentialindeterminates , a differential polynomial on U is a polynomial on U and therelative derivatives Θ U (if U = { x , x } , Θ U = { x , x (cid:48) , x (cid:48)(cid:48) , . . . , x , x (cid:48) , x (cid:48)(cid:48) , . . . } and an example of differential polynomial is x (cid:48) − x x (cid:48)(cid:48) x + x (cid:48)(cid:48) x (cid:48)(cid:48)(cid:48) x x (cid:48) ).In this paper, differential indeterminates can be considered as real functions de-pending on the single independent variable t , which we may think as the time.We also refer to differential indeterminates as dependent variables . Consideringby Q { U } the set of all the differential polynomials with rational coefficientson the variables U , it is a ring (i.e. a mathematical structure equipped withsum and multiplication and satisfying certain properties: for an introduction toalgebraic topics in the non-differential case see Lang [2005]) with a derivation,thus a differential ring .The set of all the polynomials solving some polynomial conditions is capturedby a structure named ideal . Given an ideal I of polynomials: I contains the nullpolynomial; the sum of two polynomials in I belongs to I ; the multiplication ofa polynomial in I with another polynomial (not necessarily in I ) still belongs to I . In algebraic geometry, the set of polynomials satisfied by a given polynomialsystem forms an ideal that is also radical: an ideal I is said to be radical if a ∈ I whenever there exists some p ∈ N so that a p ∈ I . In the differential case,an ideal I is a differential ideal if it is stable under derivation, which is a (cid:48) ∈ I ,for all a ∈ I . Besides, exactly as in non-differential case, a differential ideal I is radical if a p ∈ I implies a ∈ I for any integer p > x (cid:48) − x = 0 ( x is the onlydependent variable). The analytical solutions are the zero function x ( t ) = 0and the family of parabolas x ( t ) = ( t + c ) where c is an arbitrary constant.These are also solutions of all the derivatives of x (cid:48) − x = 0 (i.e. 2 x (cid:48) ( x (cid:48)(cid:48) −
2) =0 , x (cid:48) x (cid:48)(cid:48) + 2 x (cid:48)(cid:48) ( x (cid:48)(cid:48) −
2) = 0 , . . . ) and of every differential polynomial a power ofwhich is a finite linear combination of the derivatives with arbitrary differentialpolynomials as coefficients, i.e. every element of the radical of the differentialideal generated by x (cid:48) − x .When we are interested in a restriction of all the variables, we take a projec-tion of an ideal, and the operation is called elimination . The projection of anideal is still an ideal. In the elimination process for both purely algebraic anddifferential systems, we need as input a system of (differential) polynomials andan order defining the priority of the variables to be eliminated, so called ranking .The output is a system (or a family of systems when splitting is necessary) that15s equivalent to the input system restricted on some variables. Even if in prac-tice the worst case complexity of the algorithms makes problems untreatable,in principle elimination is always possible.For the differential elimination (and in general to decide the membershipof a differential polynomial in a radical differential ideal) the key algorithmis Rosenfeld-Gr¨obner one. As readable in the description of the command inMaple , given a system Σ containing differential-polynomial equations and in-equations, the algorithm splits the given system into other systems defined bycertain equations and inequations (the result and the number of cases dependon the ranking of the variables). The solutions of Σ are given by the unionof the general solutions of each of the returned systems. Every system is analgebraic structure named differential regular chain , and the radical differentialideal generated by Σ is the intersection of all the obtained differential regularchains.About the differential ranking, if U is a finite set of dependent variables, a ranking over U is a total ordering over the set Θ U of all the derivatives of theelements of U which satisfies, for all a, b ∈ Θ U , a (cid:48) > a and a > b ⇒ a (cid:48) > b (cid:48) .When U = { a } (there is a unique dependent variable), there exists only oneranking: · · · > a (cid:48)(cid:48) > a (cid:48) > a . The choice of the ranking is non-trivial when wehave more dependent variables. For our purposes, we have to introduce the orderly and eliminating rankings.A ranking is said to be orderly if, for every a, b ∈ U and for every positiveinteger value of i and j , i > j ⇒ a ( i ) > b ( j ) .If U and V are two finite sets of differential variables, one denotes U (cid:29) V every ranking so that any derivative of any element of U is greater than anyderivative of any element of V . Such rankings are said to eliminate U withrespect to V .Fixed a ranking, the leader is the highest ranking derivative appearing ina differential polynomial. Thus, given x (cid:48) − x x (cid:48)(cid:48) x + x (cid:48)(cid:48) x (cid:48)(cid:48)(cid:48) x x (cid:48) , withany orderly ranking the leader is x (cid:48)(cid:48)(cid:48) (there are no x with derivative more than1). We have the same leader with the ranking eliminating x . On the contrary,with the ranking eliminating x the leader is x (cid:48) .To sum up, recalling [Hubert, 2003, pp. 41–42], given a system of differentialpolynomials Σ in the dependent variables x , . . . , x n , with an appropriate choiceof the ranking we can: • check whether a differential polynomial is a solution of Σ = 0; • find the differential polynomials satisfied by the solutions of Σ = 0 in asubset of the dependent variables (we can obtain the equations governingthe behavior of the components x , . . . , x m , with m < n ); • find the lower order differential polynomials satisfied by the solutions ofΣ = 0 (in particular, we can inquire whether the solutions of the systemare constrained by purely algebraic equations). Cf. a lot leftto do. Even though Pritchard and Sit [2007] and the approach proposed byMarkus Rosenkranz with regard to symbolic methods for (linear) boundaryproblems (e.g. Rosenkranz et al. [2012]), at my knowledge the symbolic solutionof general initial value problems is far away from being solved. To describe the behavior defined by TMMs, we adopt the behavioral approach of mathematical models [Polderman and Willems, 1998, pp. 1–8]. The maindifference between the behavioral approach and the input/output one is that inthe first one we consider all the variables without the need of distinguishing thembetween input and output. The advantage of missing this distinction comes fromthe fact that considering interconnection between components (the so-called feedback ), it is generally hard or impossible to understand which variables areinputs and which ones are outputs. TMMs were firstly introduced adopting theinput/output approach Milici [2012a], while in this paper we use the behavioralapproach to analytically study the machines with differential algebra instead ofclassical infinitesimal calculus.A mathematical model posits that some things can happen, while otherscannot. We can formalize this idea by stating that a mathematical model se-lects a certain subset from a universum of possibilities. This subset consists ofoccurrences that the model allows, that it declares possible. We can refer to thesubset in question as the behavior of the mathematical model. Such exclusionlaws are usually expressed in terms of equations in some variables. The behav-ior obtained considering all the variables is called total . If we want to restrictonly to the some variables, we speak of restricted behavior (or simply behavior,restricted may be implicit). With this approach two machines are equivalent ifthey have the same behavior. Before deepening the exploration of TMMs, webegin to explore the behavior of algebraic machines.
Proposition 1.
The total behavior of algebraic machines with n points and m rods is a real algebraic set with integer coefficients in n + m ) variables.Proof. First of all, for algebraic machines we can consider as behavior the set ofthe configurations allowed by the constraints of the machine, i.e. the possiblecontemporary positions of the various points. Being on a plane, each of thetwo coordinates of any point has to be a real value. Given the possibility oftranslating onRod and dist in algebraic polynomials with integer coefficients(cf. section 2.3), we can consider as variables the coordinates of the n points P i .At them we have to add, for each of the m rods, the point Q j,k . Therefore, in the For example, we are interested in the following problem: given two TMMs with theirrelative initial configurations, are their behaviors equivalent? Analytically, the question arises:given two systems of differential equations with the relative initial conditions, are the systemsequivalent? We are looking for an algorithm to symbolically solve this problem. Differentialalgebra language does not permit even to express this problem because we need to explicitlystate the relation between the dependent variables and the independent one (to pose the initialcondition). R n + m ) , the behavior (the possible configurations) is the solution ofa system of polynomials with integer coefficients on 2( n + m ) variables.Note that, from an algebraic perspective, it is better to consider rationalcoefficients instead of integer ones to allow the existence of the inverse of everynon-zero coefficient (to define a so called polynomial ring ). However, multiply-ing by the minimum common multiple of all the denominators, every polynomialwith rational coefficients is equivalent to another one with only integer coeffi-cients, hence we continue considering only integer coefficients. Proposition 2.
Given any polynomial p with integer coefficients in n real vari-ables, we can consider an algebraic machine having as restricted behavior exactlythe zero set of p .Proof. Consider p on x , . . . , x n with integer coefficients, let such coefficientsbe c , . . . , c k . Introduce the n moving points P i = ( x i ,
0) by onRod(0,0,i) (keep in mind that the x-axis is r (0 , k fixed points P n + j = ( c j ,
0) by dist(0,0, c j ,n+j) . The polynomial p is made up by sums and multiplicationsof x i and c j . Thus, using the code of Problems 4 and 6, we can construct thepoint P h = ( p,
0) (with a certain index h ) in function of x , . . . , x n . To conclude,we can impose p to be constantly equal to 0 by dist(0,0,0,h) . Restricting thebehavior to the abscissae of P , . . . , P n , the proposed machine provides exactlythe sought solution of the polynomial p .Note that, once physically posed the constraints, not every configurationis always reachable given a certain initial condition. That happens becausereal algebraic sets (i.e. the solution of systems of algebraic polynomials onreal variables) are not always made up by connected components (consider thehyperbola xy − .We complete the section with the characterization of the behavior of alge-braic machines. Theorem 1 (Algebraic universality) . Considering n variables, the behaviorof algebraic machines coincides with any semi-algebraic set with integer coeffi-cients. To perform such projection one can use the
RegularChains package (out of clari-fying how to use the package, a mathematical definition of regular chains isprovided), with also the subpackage
SemiAlgebraicSetTools for the function
Pro-jection . To clarify the ideas, it follows the code to getthe behavior of the point P in the example 1 at page 7 (see Fig. 4 for the machine). roof. We split the theorem in two: 1. any machine behavior is a semi-algebraicset; 2. for any semi-algebraic set, we can consider a machine with this behavior.1. By Prop. 1, the total behavior is a real algebraic set. Thus, restrictingto n variables, the behavior has to be a semi-algebraic set.2. Any semi-algebraic set can be represented as finite union of sets eachone defined by some polynomial equations and inequalities on n variables. Forevery inequality q i ( x , . . . , x n ) >
0, we can introduce a new variable t i and thepolynomial ˜ q i ( x , . . . , x n , t i ) = q i · t i −
1. We can note that, posing ˜ q i = 0, wehave that q i (cid:54) = 0 ( q i has to divide 1) and q i ≥ q i multiplied by a non-negativevalue must give a positive value): that means that, thanks to the introductionof new variables, we can obtain the desired inequalities as projection using thenew polynomials ˜ q i .We can also consider that the system of real polynomials p = . . . = p l = 0 isequivalent to a single polynomial ( p ) + . . . + ( p l ) = 0. Hence, a semi-algebraicset can be considered as the projection of a finite union of sets each one definedby a polynomial in x , . . . , x n , t , . . . , t m : let such polynomials be P , . . . , P k .Finally, the union of such sets has to satisfy the polynomial P = P · . . . · P k ,and the semi-algebraic set has to be the projection of P = 0 on the variables x , . . . , x n . By Prop. 2, the zero-set of P can be constructed by an algebraicmachines. Thus, restricting on the variables x , . . . , x n , we got a machine withthe sought behavior. Differently from the case of algebraic machines, whose behavior is a subset of R n , the introduction of the wheel makes the previous representation ineffective.To evince it, we propose the example of a TMM: the set of its reachable points with(RegularChains): with(SemiAlgebraicSetTools) load the packages R:= PolynomialRing([x2, y2, x3, y3, x4, y4]) define the variables of our polynomialswith a given order eq1:= x2^2+y2^2=1eq2:= (x3-1)^2+y3^2=1eq3:= x2-2*x4+x3=0eq4:= y2-2*y4+y3=0eq5:= (x2-x3)^2+(y2-y3)^2=4 define the various equations proj:= Projection([eq1,eq2,eq3,eq4,eq5],2,R) compute the projection of the equations onthe last two variables considering the givenorder
Display(proj,R) show the various components of the pro-jectionThus, non considering the imaginary results and merging the singular solutions in thegeneral one, we have that the locus of the point P is given by the equation 4 x − x +(13 + 12 y ) x + ( − − y ) x + (18 y + 12 y + 1) x + ( − y − y ) x − y + 5 y + 4 y = 0.Neglecting practical computer limitation, eliminations are always theoretically computable.For example one could prove by computer algebra that the algebraic machine introduced forthe sum in problem 4 works properly without any geometrical consideration. One shouldtranslate all the instructions in algebraic equations and then consider the projection on threevariables: the two addends (the abscissae of P i and P j , i.e. x and x ) and the variable thatshould give the result of the operation (the abscissa of P [1] , denote it by t ). The projectionhas to provide an equation equivalent to x + x − t = 0. Obviously, similar reasonings areobtainable for the operations of difference and multiplication, and, suitably interpreting thegeometrical properties in analytic terms, also for all the other problems solved in the section2.4. igure 9: A simple TMM (the point P moves along a line, P [1] rotates around).Figure 10: A cycloid can be traced by a point P fixed on a rolling disk. Calling C the top ofthe disk, the tangent at P has to pass through C . is a real semi-algebraic set, so it can be obtained with an algebraic machine, butwe will intuitively see why its behavior is substantially different from the onesof algebraic machines. Example 2.
Given P = ( t, moving on the abscissa and P [0] = ( t, , consider P [1] s.t. P P [1] = 1 . In P [1] we can place a wheel so that the tangent to the curvetraced by P [1] is always in direction of P [0] (see Fig. 9). Code.
First example of a TMM that is not an algebraic machine. Consider P , P , r (0 , as usual. onRod(0,0,2) consider P on r (0 , dist(perp(0,0,2),0,1,[0]) P [0] is constrained to stay on the perpen-dicular to r (0 , passing through P , oneunit away from P dist(2,0,1,[1]) a new point P [1] is put at distance 1 from P onRod([1],0,[0]) the point P [0] is constrained to lie on therod r ([1] , wheel([1],0) pose a wheel in r ([1] , at P [1] Note that, although by construction P [0] = ( t, ± P [0] = ( t, P along the abscissa, ifthe ordinate of P [1] is strictly less than 1, it has to describe an arc of cycloid, be-cause of the geometrical property shown in Fig. 10 (for a precise analytic proofsee the section 5.1, page 28). Remind that the cycloid is a transcendental curve,thus it cannot be traced with 1 degree of freedom by algebraic machines. Onthe contrary, when P [1] assumes coordinates ( t, igure 11: For any two points ( x , y ) and ( x , y ) in the strip ] − ∞ , + ∞ [ × [ − ,
1] there is apath (the combination of the paths p , p , p ) satisfying the constraints of the machine seenin Fig. 9. It means that, given any initial position ( x , y ) of P [1] in the strip ] −∞ , + ∞ [ × [ − , x , y ) in the strip can be reached by P [1] : call ( x ∗ , x ∗ ,
1) the first apex (going from left to right) of the cycloid starting respec-tively in ( x , y ) and ( x , y ). As visible in Fig. 11, P [1] can reach ( x , y ) from( x , y ) decomposing the motion in three parts: first, P [1] reaches ( x ∗ ,
1) (it ispossible because they are on the same branch of cycloid); second, P [1] reaches( x ∗ ,
1) (they are on the horizontal line y = 1); third, P [1] reaches ( x , y ) (be-cause they are on the same branch of cycloid).So, restricting the behavior to the coordinates ( x, y ) of P [1] , the space of thereachable configurations is exactly the strip ] −∞ , + ∞ [ × [ − , P [1] can walk only certain trajectories. Thus, we cannot consider a subset of R n as universum for TMMs; we need something else. In particular, consideringthis example, the universum of a TMM can be made up by a (generally infinite)set of curves satisfying both the configuration conditions of the holonomic con-straints and the path conditions imposed by non-holonomic ones. Let’s defineit more precisely.Differently from the synthetic approach, differential geometry introducescalculus for the investigation of curves. In particular, curves are represented in aparametrized form as a class of equivalence on vector-valued functions . Comingback to TMMs, we can continue the interpretation of variables as coordinatesof specific points of machines as done in algebraic ones. But, unlike before, it isno longer enough to consider variable as real numbers, but, to introduce pathconstraints, we can consider these variables as real functions ( R → R ), where A parametric curve γ is a vector-valued function I → R n (where I is a non-empty intervalof real numbers) of class C r (i.e. γ is r times continuously differentiable, eventually also ∞ times differentiable). If we consider t ∈ I , t is the parameter of γ , and γ ( I ) is the image ofthe curve (considering t as time, γ ( t ) represents the trajectory of a moving particle).The same image γ ( I ) can be described by several different C r parametric curves: the aimof differential geometry is to study the curve independently from reparametrizations . In doingthat, we can consider curves as an equivalence class on the set of parametric curves. Theequivalence class is called a C r curve (equivalent C r curves have the same image). For adetailed discussion, see, for example, Do Carmo [1976]. C ∞ , i.e. smooth functions.With reference to the example of the machine in Fig. 9, we need to consideras universum something like manifolds of curves. However, curves can be definedas classes of equivalence over vector-valued functions. So, to mathematicallysimplify the definition, we suggest to consider a “manifold of C ∞ functions”as universum for TMMs. In particular, considering n variables, these functionshave to be R → R n .Algebraic machines are a restriction of TMMs, so we can observe how the in-terpretation of the universum/behavior as real semi-algebraic set is reformulatedas manifold of functions. From the point of view of paths, algebraic machinesallow any path moving inside the defined semi-algebraic set S ⊂ R n , so themanifold of functions has to be made up by all the functions of class C ∞ havingtheir image inside S . We have just defined a manifold of smooth functions as universum of a TMM.Variables are coordinates of specific points, and are considered as functions. Asintroduced in section 2.3, both wheel constraints and algebraic conditions aretranslatable in polynomials in the variables and their derivatives: as seen in thealgebraic part (section 3), such polynomials are named differential polynomials and constitute the basis for differential algebra. More formally:
Proposition 3.
The total behavior of a TMM with n points and m rods is themanifold of all the smooth real functions R → R n + m ) satisfying a system Σ ofdifferential polynomial equations with integer coefficients.Proof. We just have to translate all the instructions defining the machine in dif-ferential equations (purely algebraic equations if the instruction is not wheel ),taking care of introducing the auxiliary points Q i,j when introducing a rod. Byconstruction, all the coefficients has to be integers. Renaming x , . . . , x n + m ) allthe coordinates in function of the time t , and considering p , . . . , p l the obtaineddifferential polynomials on x , . . . , x n + m ) , the total behavior is { ( x , . . . , x n + m ) ) | x i : R → R , x i ∈ C ∞ , p = . . . = p l = 0 } .Note that we found an analytical form only to the total behavior: for therestricted behavior, in general we have to eliminate the unwanted variables. Proposition 4.
Given a system Σ of differential polynomial equations withinteger coefficients, we can construct a machine having as restricted behaviorthe manifold of the solutions of Σ .Proof. First of all, we can convert Σ in an equivalent system involving morevariables but with only first derivatives. Let y , . . . , y m be the variables of Σ,and let k i be the maximum derivative of y i present in Σ (i.e. y ( k i ) i appears, but y ( k i +1) i doesn’t): for every variable we introduce new auxiliary variables y i,j (for j = 1 , . . . , k i ) and the differential polynomials y i, − y i , y i, − y (cid:48) i, , . . . , y i,k i − y (cid:48) i,k i − . (1)22 igure 12: Construction of the derivative of the variables x i , x j . Adding such new polynomials and modifying the system by substituting y ki with y i,k , we get an equivalent system involving only purely algebraic polynomialsand the first order derivatives of (1). To simplify the notation and use a singleindex (still involving only first order derivatives), denote the various variables y i,k (for all i and k ) by x , . . . , x n .Working on real values, we can convert the system Σ given by p = . . . = p l =0 in a single differential polynomial p = ( p ) + . . . + ( p l ) = 0. This differentialpolynomial is a polynomial on x , . . . , x n and their first derivatives. As seenin Prop. 2, we can solve by algebraic machines any polynomial, hence we justhave to show how to construct the derivatives of the variables x , . . . , x n (thisconstruction was firstly expressed in Milici [2012a] to solve polynomial Cauchyproblems). For this purpose we introduce the following code. Code.
Given the fixed points P , P and r (0 , as abscissa, in this code weintroduce the following points: P = ( t, , P [0] = ( t + 1 , , P [ i ] = ( t, x i ) , P [ i + n ] =( t + 1 , x i + x (cid:48) i ) (for i = 1 , . . . , n ). nRod(0,0,2) consider P on r (0 , dist([0],0,0,sum(1,2)) denoting P = ( t, , P [0] is constrained tostay in ( t + 1 , onRod(perp(0,0,2),0,[1]) . . . onRod(perp(0,0,2),0,[n]) each P [ i ] is constrained to lie on the rodperpendicular to r (0 , passing through P onRod(perp(0,0,[0]),0,[n+1]) . . . onRod(perp(0,0,[0]),0,[2n]) each P [ n + i ] has to lie on the rod perpendic-ular to r (0 , passing through P [0] onRod([1],0,[n+1]) . . . onRod([n],0,[2n]) each rod r ([ i ] , has to pass through P [ n + i ] wheel([1],0) . . . wheel([n],0) in these lines we introduce wheels in r ([ i ] , P = ( t,
0) by a cart on the abscissa( t can assume any real value). Note that t is arbitrary, the important thingis that all the various x i are considered in correspondence of the same t : t can be viewed as the independent variable in function of which the variousfunctions (dependent variables) are computed. Then, consider the points P [1] =( t, x ) , . . . , P [ n ] = ( t, x n ). On these points, we can put n rods: call r i = r ([ i ] , P [ i ] . Put also a wheel on every r i in correspondence of P [ i ] .We can construct the rod of equation x = t + 1: call x ∗ i the ordinate of the point P [ n + i ] in the intersection of x = t + 1 and r i .For what has been observed about the role of the wheel, r i has to be tan-gent to the graph of ( t, x i ), hence x ∗ i will be x i + x (cid:48) i . Obviously, it was notstrictly necessary to construct the rod of equation x = t + 1: in the case ofa rod of equation x = t + a (for any constant a (cid:54) = 0), the intersection of r i with the new rod is ( t + a, x i + ax (cid:48) i ) (in other words, x ∗ i = x i + ax (cid:48) i ). It meansthat we can construct the point ( x (cid:48) i ,
0) that can be used as a new variable, andso the differential polynomial can be considered as a purely algebraic polyno-mial on x , . . . , x n , x (cid:48) , . . . , x (cid:48) n . Specifically, the points ( x (cid:48) i ,
0) are obtainable by diff(inv([n+i]),inv([i])) (with i = 1 , . . . , n ).So, the possibility of solving polynomials with algebraic machines assuresthat, for every system of differential polynomial equations Σ, we can considera TMM having as restricted behavior (restricted to the original y , . . . , y m ) thesolution of Σ. As a first example of passage from differential equation to TMM, we can considerthe problem y (cid:48) = y . To construct a machine solving it we can start consideringa cart ( t,
0) on a fixed rod (that we consider as abscissa), a rod perpendicularto the abscissa and translating according to the value of t , and on this rod thepoint ( t, y ). As already observed, instead of the rod of equation x = t + 1, wecan consider any other form x = t + a . In particular, it is simpler if we adopt a = −
1. Thus, y ∗ = y + ay (cid:48) = 0 (for the problem is y (cid:48) = y ). Therefore, we haveto introduce the rod r passing through ( t, y ) and ( t − , igure 13: A machine solving the differential equation y (cid:48) = y (left) and the relative slope field(right). it in correspondence of ( t, y ), obtaining the machine of Fig. 13. Conceptually,this machine is constructively using the property of the exponential curve ofhaving a fixed-length subtangent (i.e. the segment connecting ( t − ,
0) and( t, Example 3.
Consider the machine defined by P = ( t, y ) and the constraintthat the direction of the point ( t, y ) has to be the line passing through ( t − , . Code.
Consider the fixed points P , P . onRod(2,0,diff(2,1)) consider P = ( t, y ) , thus ( t − , is constrainedto lie on the rod r (2 , wheel(2,0) pose a wheel in r (2 , at P Because of the wheel, ( t (cid:48) , y (cid:48) ) has to be parallel to r (2 , t (cid:48) y − y (cid:48) = 0.Note that the obtained differential equation is different from the original one( y (cid:48) − y = 0). The difference is given by the implicit assumption that t (cid:48) = 1.When we construct a machine solving a system of differential equations withthe method seen in Prop. 4, we implicitly assume that t is the independentvariable, so everything is obtained in function of its value. The introduction ofa new variable (with constant derivative 1) for the independent one is a standardmethod to pass from a differential polynomial involving also the independentvariable to an equivalent polynomial not depending directly on the independentvariable.In summary, given a system of differential polynomials Σ, with the methodof Prop. 4 we can construct a machine solving it, but the system Σ ∗ obtainedanalyzing this machine is slightly different from the original Σ. If we wantto obtain Σ from Σ ∗ , we have to add the condition x (cid:48) i = 1 for the variable x i that represents the abscissa of the independent point ( t, x (cid:48) i = 1 it follows x i = t + k , i.e. x i is exactly the independent variable eventuallytranslated of a constant k ). 25 .4 Note on initial conditions The total behavior of TMMs can be analytically defined by a system of differ-ential polynomials. However, when a machine is considered to work on a plane,the initial position of its components can be considered implementing the initialconditions. In this section, we focus on how to apply these initial conditions.Physical realizations of TMMs are devices that can be lifted and downed onthe plane. While the device is not yet downed on the plane, there are fewerworking constraints (because of the lack of wheel friction), so we can movesome points that lose some degrees of freedom when wheels touch the plane.Therefore, if we consider TMMs as physical devices, their assembly and use canbe distinguished in two different steps:1. composition: the various parts are assembled in order to construct themachine;2. friction on the plane: the machine is “put on the plane,” so wheels avoidlateral motions.The difference between these two steps is the role of the wheel. In the first casethe machine is constructed but, considering it lifted from the plane, the wheelconstraints do not work, so on the machine only the holonomic constraints areactive (the ones of algebraic machines). When we ideally put the constructedmachine on the plane, wheels begin to have friction on the plane, and conse-quently the related nonholonomic constraints begin to work.While the composed machine is already defining differential polynomial equa-tions, the activation of the friction is related to the posing of initial conditions.In fact, in the instant when the constructed machine touches the plane (andthe wheel friction begins), all the points have a certain position: the values ofthe variables relative to these positions can be viewed analytically as the initialconditions. Therefore, to pose an initial condition to some variables, we haveto suitably move the points (the position of which is related to the wanted vari-ables) when the device is lifted. The downing of the device assures that thevariables solve the Cauchy problem.To clarify these ideas, as an example, we propose a machine solving thedifferential equation − y (cid:48)(cid:48) ( t ) = y ( t ). According to different initial conditions,the same machine can generate the sine (posing y (0) = 0 , y (cid:48) (0) = 1) and thecosine function (with initial conditions y (0) = 1 , y (cid:48) (0) = 0).As seen in Fig. 12, once introduced the point ( t, y ( t )), we can constructthe point ( t + 1 , y ( t ) + y (cid:48) ( t )). Reporting the length − y (cid:48) ( t ) as represented inthe Fig. 14 (parallel dotted lines represent the translation of lengths, withoutvisualizing all the necessary components), we can construct ( t, − y (cid:48) ( t )). Then,once constructed ( t + 1 , − y (cid:48) ( t ) − y (cid:48)(cid:48) ( t )), it is possible to impose y ( t ) = − y (cid:48)(cid:48) ( t )by reporting the length − y (cid:48)(cid:48) ( t ).Now it is time to impose initial conditions. For cosine requires totally similarsteps, let’s consider only the sine function, thus we have to impose y (0) =0 , y (cid:48) (0) = 1. Physically, this condition has to be posed after the construction ofthe machine, and before the “activation” of the friction of the wheels. First, wemove the cart in ( t,
0) until it reaches the position (0 , t, y ( t )) and ( t, − y (cid:48) ( t )) until they (respectively) reach the positions (0 ,
0) and(0 , − igure 14: Sketch of a machine for y = − y (cid:48)(cid:48) . Parallel dotted lines represent the translationof lengths. (ideally: when the machine is not yet put on the plane), the nonholonomicconstraints of wheels can be activated (the machine can be finally put on theplane, allowing the friction of the wheels on the plane). In this way the machinegenerates exactly the sine function.In contrast to the case of the exponential, the sine function is constructedusing a second order differential equation, so it is not possible to consider thewheel solving a static graphical slope field. Indeed, the slope of the rod with awheel is dynamically defined in function of the position of the other wheel. With the introduction of tractional motion machines, we can overcome Carte-sian geometry still relying on the idealization of suitable machines, and, thanksto differential algebra, we can also provide a well-defined language and set ofalgorithms for the analytical counterpart without the need of the infinity orof approximations (as the mathematical concept of limits). In this section, wesuggest some applications of differential algebra for such machines.According to [Bos, 2001, p. 287], Descartes’ geometric problem solvingmethod consisted of an analytic part (using algebra to reduce any problem to anappropriate equation) and a synthetic part (finding the appropriate geometricconstruction of the problem on the basis of the equation). Considering analysisby differential algebra and synthesis by TMMs, the same Cartesian problem-solving method can be extended beyond algebraic boundaries by the followingsteps:1. start from a problem about TMMs,2. convert it in differential polynomials,27. solve the problem with differential algebra algorithms,4. when requested, after the simplification, find the specific solution withdiagrammatic construction of TMMs.Regarding the third step, we suggest to manipulate equations with the
Differen-tialAlgebra package of the computer algebra system Maple, of which we includecommands in footnotes.
As a first example, by automated reasoning we prove what has been informallyobserved in the section 4.1 about the behavior of the machine of Ex. 2 (page20).Consider P = ( t,
0) and P [1] = ( x, y ) moving around P at unitary distance,so ( x − t ) + y = 1 . (2)The wheel in P [1] has as direction P [1] − P [0] , i.e. P (cid:48) [1] = ( x (cid:48) , y (cid:48) ) has to be parallelto P [1] − P [0] = ( x − t, y − y (cid:48) ( x − t ) = x (cid:48) ( y − . (3)Thus, we have two equations (the first purely algebraic and the second dif-ferential) in t, x, y . If we are interested in the curve traced by P [1] , we can usedifferential elimination to eliminate t . We can proceed with the following steps:1. consider the differential ring R with rational coefficients having as variables t, x, y , and adopt a ranking eliminating t ;2. consider the ideal I in R generated by the two differential polynomials;3. consider in I the differential regular chains eliminating t .We can translate these steps in commands for computer algebra software . Inparticular we obtain that the differential regular chains (for the ideal generated In Maple we can perform these operations with the following code lines (commented onthe right): with(DifferentialAlgebra) load the package
R := DifferentialRing(blocks=[t,x,y],derivations=[a]) construct the differential ring with as independentvariable a , and dependent ones t, x, y with theranking t (cid:29) x (cid:29) y p := (x(a)-t(a))^2+y(a)^2 = 1 p is an algebraic equation q := (diff(y(a), a))*(x(a)-t(a)) = (diff(x(a),a))*(y(a)-1) q is a differential equation ( diff(f(a), a) standsfor the derivative df/da ) ideal := RosenfeldGroebner([p, q], R) ideal is the radical differential ideal generated by p , q Equations(ideal) returns the equations of idealInequations(ideal) returns the inequations of ideal
Note that the commands
Equations(ideal) and
Inequations(ideal) show the differentialregular chains for the ideal in t, x, y .Once obtained the differential regular chains reduced with respect to a certain ranking, theelimination of the greater depending variable only consists in taking all and only the equationsand inequalities of the differential regular chains where the variable and its derivatives do notoccur. Using Maple, it can be achieved with the command:
Equations(ideal, leader 28y the two equations characterizing the TMMs) reduced with the ranking t (cid:29) x (cid:29) y are: C = { ty (cid:48) + x (cid:48) y − x (cid:48) − xy (cid:48) = 0 , x (cid:48) y − x (cid:48) + y (cid:48) y + y (cid:48) = 0 , y (cid:48) (cid:54) = 0 , x (cid:48) y − x (cid:48) (cid:54) = 0 , y − (cid:54) = 0 } ; C = { t − x = 0 , y − } ; C = { t − xt + x + y − x (cid:48) = 0 , y (cid:48) = 0 , t − x (cid:54) = 0 } ; C = { t − x = 0 , x (cid:48) = 0 , y − , y (cid:54) = 0 } . But, as said, we are not interested in the behavior of t , so, if we eliminate it(considering the given ranking of the variables, we can just skip all the equationswith t in C , C , C , C ), we obtain C ∗ = { x (cid:48) y − x (cid:48) + y (cid:48) y + y (cid:48) = 0 , y (cid:48) (cid:54) = 0 , x (cid:48) y − x (cid:48) (cid:54) = 0 , y − (cid:54) = 0 } ; C ∗ = { y − } ; C ∗ = { x (cid:48) = 0 , y (cid:48) = 0 } ; C ∗ = { x (cid:48) = 0 , y − , y (cid:54) = 0 } . Even though it is possible to do some simplifications, we adopted the givenform (that is exactly the one given by the Maple code) to evince the fact thatany reasoning can be conducted in a purely formal way without consideringthe semantic meaning. We can observe that C ∗ and C ∗ does not provide usanything interesting but single points.On the contrary, we can observe that C ∗ contains as equation the generalsolution which, rewritten as an ODE, becomes the differential equation of thecycloid: (cid:18) dydx (cid:19) = 1 − y y . Another solution out of arcs of cycloids is made up by the line y = 1, that wasexcluded in C ∗ because of its inequalities.We can also ask ourself which constraints can be added to construct exactlya cycloid. According to the property seen in Fig. 10 (page 20), we can impose anew tangent condition in another point of the circumference of the rolling disk.As such a point, we can consider the point symmetric to P [1] with respect tothe center P , as visible in Fig. 15. Such machine can be defined by appendingfew instructions to the code of the example 2: Code. We add a new point P [2] and pose a wheel on it. dist(2,0,-1,[2]) the point P [2] is introduced as the symmet-ric of P [1] with respect to P onRod([2],0,[0]) the point P [0] is constrained to lie on therod r ([2] , wheel([2],0) pose a wheel in r ([2] , at P [2] The wheel in P [2] = (2 t − x, − y ) has as direction P [2] − P [0] . That meansthat P (cid:48) [2] = (2 t (cid:48) − x (cid:48) , − y (cid:48) ) has to be parallel to P [2] − P [0] = ( t − x, − y − t (cid:48) − x (cid:48) )( − y − 1) = − y (cid:48) ( t − x ) . (4)29 igure 15: A machine for the cycloid. We introduce a new point P [2] symmetric of P [1] withrespect to P . Also in P [2] we pose a wheel on the rod passing through P [0] . If we consider the ideal generated by the three equations (2), (3) and (4), wecan compute the relative differential regular chains eliminating t . We obtainthat x and y have to satisfy the differential systems C ∗∗ , C ∗∗ , C ∗∗ where C ∗∗ = C ∗ , C ∗∗ = C ∗ , C ∗∗ = C ∗ , i.e. there is no longer the solution y = 1 given by C ∗ .Thus, with the new conditions, the point ( x, y ) is always constrained to walkalong a cycloid. It is time to give a characterization theorem for the behavior of TMMs, somehowthe extension of Kempe’s (algebraic) universality theorem to the differentialcase. Theorem 2 (Differential universality) . Considering n variables, the behavior ofa TMM coincides with the union of the solutions of a finite number of systemsof differential polynomial equations and inequations with integer coefficients andwith real functions as independent variables.Proof. We split the theorem in two: 1. the machine behavior is the union of thesystems; 2. for any finite set of systems, we can consider a machine with thisbehavior.1. By Prop. 3, the total behavior of any TMM is the solution of a differentialpolynomial system Σ in m variables ( m ≥ n ). Thus, to restrict to n variables,we can use the Rosenfeld-Gr¨obner algorithm of section 3, hence the behaviorcan be given as a finite union of systems of differential-polynomial equationsand inequations.2. With certain modifications, we can reuse some ideas in the proof of the-orem 1 (algebraic universality): for every inequation q i ( x , . . . , x n ) (cid:54) = 0 we canintroduce a new variable t i and the differential polynomial ˜ q i ( x , . . . , x n , t i ) = q i · t i − 1. Thus, even though with more variables, we got that every system canbe considered as made up by only differential polynomial equations with integercoefficients. Then, we can use the peculiarity of the variables of TMMs to bereal functions: any system of real differential polynomials p = . . . = p l = 0 is30quivalent to a single polynomial ( p ) + . . . + ( p l ) = 0. Let P , . . . , P k be thedifferential polynomials identifying the k systems: the union of their zero setshas to satisfy the polynomial P = P · . . . · P k . By Prop. 4, the zero-set of P can be constructed by a TMM. Thus, restricting on the variables x , . . . , x n , wegot a machine with the sought behavior.While theorems of algebraic universality and differential universality havemany similarities, we have to highlight that in the algebraic case we have in-equalities while in the differential only inequations. Remind that differentialalgebra does not distinguish between real or complex or other kind of indeter-minates (Rosenfeld-Gr¨obner algorithms works for any differential ring), whilesemi-algebraic sets are defined specifically for the real case.We can also define the nature of the functions that TMMs can generate.After the definition of differentially algebraic functions [Rubel, 1989, p. 777],we use it to give a classification of the constructible functions. This resultis particularly interesting from an historical perspective: differently from thealgebraic case, the classification of the curves traced by tractional motion waspreviously missing. Definition 1. A function y is differentially algebraic (shortly: D.A.) if it sat-isfies an algebraic differential equation, i.e. a differential equation in the form P ( t, y, y (cid:48) , . . . , y ( n ) ) where P is a nontrivial polynomial in n + 2 variables. The non-triviality condition is essential because every function is solution of0 = 0. Proposition 5 (constructible functions) . The curves generated by a TMM areall and only the image of D.A. functions.Proof. Given a certain TMM, consider a point ( x, y ) of the machine with onedegree of freedom (if the point has two or more degrees of freedom it does notdefine a curve). The curves traced by this point are defined as all the value of x, y satisfying the restricted behavior, i.e. many systems of differential polynomialequations and inequations in x, y . Now we can be interested in interpreting acurve as the graph of a function (at least locally). So, we can consider the curveas a function y = f ( x ) [respectively x = g ( y ) in a neighborhood of verticaltangents; however, we will no longer consider this case because we only haveto switch the role of x and y ]. To achieve this aim, we can no longer consider x as a dependent variable, but as an independent one. Algebraically, this istranslated (as seen in the section 4.3) by the “independentization condition” x (cid:48) = 1. Indeed, if we add the new condition to the systems in x, y , we canagain consider the elimination of x obtaining a family of differential regularchains only in y . Thus, we find that the curves ( t, y ( t )) are (locally) solutionsof differential polynomials in y , so D.A.Conversely, we have to recall that any D.A. function satisfies an algebraicdifferential equation with integer coefficients [Rubel, 1989, Th. 1, p. 778].As observed in section 4.3, the introduction of a new variable with constantderivative 1 for the independent one allows the construction of an equivalentpolynomial not depending directly on the independent variable. Once eliminatedthe new variable, the obtained differential polynomial in y can be solved by aTMM. Hence, the constructible functions are all and only the differentiallyalgebraic ones. 31his result is important because it means that TMMs generate a new dualismbeyond algebraic/transcendental (and this time about functions, not curves oralgebraic varieties as done with algebraic machines). Note, however, that amachine can construct functions that globally are not D.A., as visible sincethe first introduction of TMMs (Milici [2012a] concerned the construction of amachine tracing a cycloid, that globally is not D.A.), but locally each of thesefunctions has to be D.A.All the elementary functions are D.A., and even most of the transcendentalfunctions that we find in analysis handbooks. Historically, the first example ofnon-D.A. function was the Γ of Euler, as proven in H¨older [1886]. Note that Γfunction is not even locally D.A., that is why it cannot be constructed by TMMs.As an example, we can continue with the cycloid, observing some differenceswhen we “independentize” different variables.Adding the constraint x (cid:48) = 1 to the equations (2), (3) and (4), we canconsider y in function of x . This time, with a ranking eliminating t and x , weobtain only one regular chain: C { x (cid:48) =1 } = { y (cid:48) y + y (cid:48) + y − y (cid:48) y + y (cid:48) (cid:54) = 0; y + 1 (cid:54) = 0 } . This representation is not useful to identify the traced curve as the usualparametrization of a cycloid. This identification is more visible if we “indepen-dentize” another variable. Consider the additional constraint t (cid:48) = 1 insteadof x (cid:48) = 1. Even in this case, we obtain only one regular chain that, uponeliminating t , becomes: C { t (cid:48) =1 } = { x (cid:48) − y − y (cid:48) + y − y (cid:48) (cid:54) = 0 } Now we can observe that this representation is the one of (cid:40) x = t + cos ty = − sin t Indeed, instead of the trigonometric functions we can convert the system in apurely differential polynomial one: x (cid:48) = 1 + yy (cid:48) + y = 1 y (cid:48)(cid:48) = − y Computationally, we can check that it has as regular chains exactly C { t (cid:48) =1 } (inboth cases the computed regular chain is { y − x (cid:48) + 1 = 0; x (cid:48)(cid:48) + x (cid:48) − x (cid:48) =0; x (cid:48)(cid:48) (cid:54) = 0 } ).Obviously different machines can construct the same manifold of zeros. Re-maining on the example of the cycloid, we can construct a TMM having a pointof coordinate ( x, y ) satisfying the equations of C { t (cid:48) =1 } by the standard methodseen in Prop. 4 and Fig. 12. This way, we consider separately the variables x and y by the introduction of the points ( t, x ) and ( t, y ), impose algebraicand differential conditions on both dependent variables, and then we constructthe point having as coordinate ( x, y ). This method is general, but of course,does not provide the simplest machine (we do not intend to study the notion ofsimplicity, consider “simplest machine” in an intuitive sense).32 .3 Equivalence between TMMs In the previous example, we have seen that two radical ideals were equivalentbecause they had the same representation. However, the opposite in generaldoes not hold.As seen in the section 4.2, the total behavior of TMMs is the solution of asystem of differential polynomial equations, so the restricted behavior is the re-striction to the relative manifold of solutions on some variables. Before checkingthe equality test between two machines on certain variables, we have to supposethat the variables of the restricted behavior are in the same number in the twomanifolds.Let X = { x , . . . , x n } , Y = { y , . . . , y m } , Z = { z , . . . , z l } be sets of de-pendent indeterminates. Consider the radical differential ideal A generated bythe differential polynomials p , . . . , p i on X ∪ Y , and consider B generated by q , . . . , q j on X ∪ Z .In general, there is no known method to check the equality of two radicaldifferential ideals represented by regular differential chains (it is related withthe so called Ritt’s open problem Golubitsky et al. [2009]), but in this case wehave much stronger hypothesis. In fact, we known the generators of the totalbehavior: with this condition the solution is easily achievable and computable.To check the equality between two total behaviors (i.e. between radical dif-ferential ideals given by a finite set of generators), we can fix a certain rankingand compute the regular differential chains using the Rosenfeld-Gr¨obner algo-rithm, and then we can test whether all the generators of the first ideal belongto the second and vice-versa .But we are interested in behaviors obtained by eliminating some variables,that are in general represented by an intersection of families of regular chains,and there is no known algorithm to pass from a representation of families ofregular chains to a list of generators.Given the ranking Z (cid:29) Y (cid:29) X (or Y (cid:29) Z (cid:29) X ), compute the fami-lies of regular chains R A and R B representing respectively A and B (e.g usingRosenfeld-Gr¨obner’s algorithm). Let R ∗ A be R A without the equations/inequationsinvolving Y and, similarly, let R ∗ B be R B eliminating Z . We can verify the in-clusion of A in B restricted to X by checking whether all the p i belongs to R ∗ B ;similarly for the opposite inclusion check whether q i belongs to R ∗ A . If both theinclusions are verified, the two systems are equivalent restricted to the variables X . Note that we have treated TMMs without any reference to initial conditions.As far as my knowledge goes, the equality problem is still open if we introduceinitial values (cf. note 4 at page 17). With regard to some positive results, wecan consider Buchberger and Rosenkranz [2012], which provides an algorithmfor the symbolic solution of linear boundary problems, passing from differential According to the notation of this section, in the case of total behaviors Y and Z are emptysets of variables. We can test whether the ideals A and B are equal using the Maple command BelongsTo of the DifferentialAlgebra package. Once given any ranking, and constructed with RosenfeldGroebner the ideals A and B by their generators, to check the equality we onlyhave to test whether all the generators of A belongs to B and vice-versa. In Maple, thecommand BelongsTo([ p , . . . , p n ],B) produces as output a list of n true/false, the i -th ofwhich indicates whether p i belongs or not to B . Conversely BelongsTo([ q , . . . , q n ],A) canbe used to check the belonging to A . igure 16: Sketch of a machine with the tangent in ( x, y ) perpendicular to the line passingthrough ( x + 1 / , x, 0) is shown. algebra to integro-differential algebras (Green’s operators).Even though without any final answer about initial conditions problems,we want to underline that differential algebra permits to check the equivalencebetween TMMs. On the other side, even in the more concrete approach tocalculus, computable analysis Weihrauch [2000], it is not possible to check theequality test between any general couple of generated objects (i.e. computablenumbers). Considering intuitively “exactness” as the property of a computa-tional frameworks (both in analytic or geometric paradigm) to be independentfrom non-finitary procedures (as unlimited approximations), we think that itwill be interesting in future to deepen the relation between the computabilityof the equality test in a theory and the exactness of the theory. Consider a TMM defining the motion of a point P of coordinates ( x, y ) so thatthe tangent in P is perpendicular to the line passing through P and the point( x + 1 / , 0) as the Fig. 16 shows. This machine is defined by the differentialpolynomial x (cid:48) − yy (cid:48) , which is the total derivative of x − y . Therefore, forevery constant c ∈ R , solutions are parabolas satisfying x = y + c . That meansthat we are able to trace any of the solutions of this TMM with an algebraicone. Hence, the general question arises: can we characterize the TMMs havingsolution constructible with algebraic machines (by eventually adding a finitenumber of real constants of integration)?Given a differential system or even its restriction on some variables, to findthe algebraic constraints satisfied we can simply use the orderly ranking in theRosenfeld-Gr¨obner algorithm. There are algebraic constraints if and only if inthe obtained family of regular differential chains there are polynomial equationswithout any proper derivative (i.e. of order 0) . In Maple, given the dependent variables x , . . . , x n and the independent variable t ,one can construct a differential ring with the orderly ranking by the command R :=DifferentialRing(blocks = [[ x , . . . , x n ]], derivations = [t]) (the double square brack-ets [[ . . . ]] indicate the orderly ranking). After the usual construction of the ideal ideal with the Rosenfeld-Gr¨obner algorithm, the purely algebraic constraints are given by Equations(ideal, order=0) . 34t is more complicated if we are interested not only in algebraic constraints,but also on first integrals given by algebraic constraints. Given a system Σof ODEs, a first integral is a function f ( t ) whose value is constant over theindependent variable t along every solution of Σ. To find such first integral,we can use the algorithm findAllFirstIntegrals proposed in Boulier and Lemaire[2015]. The algorithm takes as input a family of systems of differential polyno-mial equations and inequations, and a set of monomials µ i in the variables andtheir derivatives ( i = 1 , . . . , n ); it returns as output the coefficients α i s.t. thedifferential polynomials (cid:80) ni =1 α i µ i are first integrals of Σ.If in the algorithm we consider as input the behavior of a TMM (a familyof regular chains) and as µ i the various combination of the variables (not theirderivatives), we can obtain all the algebraic first integrals of any fixed degree(e.g. given the variables x, y , to check all the algebraic first integrals up tosecond degree we can consider as µ i the monomials x, y, x , xy, y ). However,while for any degree we can suitably consider the monomials, at my knowledgethere is no general method to check whether the behavior allows any algebraicfirst integral (of arbitrarily high degree). We considered TMMs working on a plane, but what about the extension of thesemachines beyond the planar behavior? Here we provide a sketch for the possiblephysical implementation of a 3D TMM and some shallow explanations about itsbehavior. Physically, we can introduce a cube of gelatinous material: on it wecan hypothesize that thin rods can freely move, while a small disk of center C has to locally represent the tangent plane to the surface walked by C . Similarlyto 2D TMMs, the main idea is to set some constraints to the tangent: in thiscase consider u ( x, y ) (from now on we implicitly assume the dependence on x, y )as the function to be found and, as usual in partial differential equations, let u x = ∂u∂x , u y = ∂u∂y . Considering the point C = ( x, y, u ), the disk centered in C has two perpendicular rods on the tangent plane passing respectively through( x + 1 , y, u + u x ) and ( x, y + 1 , u + u y ), as visible in Fig. 17. That imposes thesuitable partial derivative conditions u x , u y , and the lengths u x , u y can be usedto set other conditions (as in 2D TMMs).The study of such machines could be interesting as a future perspective.However, even if it is far away from the main aims of this work, we give an in-formal justification about the idea that 3D TMMs do not generate any function R → R out of the ones constructible by 2D TMMs (it is not a complete proof,we do not deal with coefficients or with a sufficiently deepened characterizationof 3D TMMs).Looking for a characterization of the behavior of such machines, consideringas variables smooth R → R functions, we can perform their sum, multiplicationand derivation. Differently from the 2D case, this time the derivation is withrespect to two different independent variables ( x, y ). Even with these machines,the suitable analytical tool is differential algebra, but with partial derivatives:such development is present since Ritt’s works [Ritt, 1950, Ch. IX]. The elimi-nation is still available, so a constructible function u : R → R defined by a 3DTMM has to be solution of non-trivial differential polynomials in u and a finitenumber of its partial derivatives.To compare constructible functions between 3D and 2D TMMs, we have35 igure 17: Sketch of 3D TMMs. To set partial derivative conditions on u ( x, y ) we can considera jelly cube. Inside, a small disk centered in ( x, y, u ) is constrained by two rods to lie on theplane passing through ( x + 1 , y, u + u x ) and ( x, y + 1 , u + u y ). That imposes u to satisfycertain values of u x and u y . to restrict somehow the functions of the 3D case. Specifically, the graph of aunary function can be obtained intersecting the graph G of ( x, y, u ( x, y )) witha plane α perpendicular to OXY . The intersection between α and OXY is aline: let its natural parametrization be x = a √ a + b t + a , y = b √ a + b t + b (asnatural parametrization we mean that the derivative of the parametrization hasunary length). Thus, called F the real function whose graph is defined by theintersection of α and G , F ( t ) = u (cid:16) a √ a + b t + a , b √ a + b t + b (cid:17) . We want toshow that F is a D.A. function.Consider the function f ( x, y ) = u (cid:16) a √ a + b x + a , b √ a + b x + b (cid:17) ( y is notused in its computation). We cannot convert such f directly with differentialpolynomials in partial differential algebra (differential algebra doesn’t deal witharguments of functions), but we can compute the partial derivatives of f . Triv-ially f y = 0, and, using the chain rule for the derivation of compositions involv-ing multivariable functions, f x = u x ddx (cid:16) a √ a + b x + a (cid:17) + u y ddx (cid:16) b √ a + b x + b (cid:17) = au x + bu y √ a + b . Thus, we can finally add the differential polynomial conditions for f x , f y to the polynomials defining the total behavior of the 3D TMM. Bythe elimination of all the dependent variables out of f , we get polynomialson f and its partial derivatives. But, being f y = 0, we can simply rename f ( x, y ) = F ( x ) , f x ( x, y ) = F (cid:48) ( x ) , f xx ( x, y ) = F (cid:48)(cid:48) ( x ) , . . . , thus obtaining a differ-ential polynomial just on F and its ordinary derivatives, i.e. F has to be a D.A.function. The richness of Cartesian setting depends on the correspondence between ob-jects of the analytical and the synthetic part. From this perspective, the roleof suitable ideal machines was central. The balance between machines, algebra,and geometry as suggested by Descartes was historically broken by the increasein importance of the analytical part with respect to geometric constructions.36n particular, infinitesimal analysis introduced infinitary tools in the analyticalpart such as series or infinitesimal elements. However, even though with somecenturies of delay, we can consider the finite approach to calculus objects of dif-ferential algebra as a legitimate descendant of polynomial algebra. Contrarily,the synthetic part can be managed with the proposed TMMs, which, as a well-defined model for tractional constructions, can be considered as an extensionof Descartes’ machines. The surprising result is that these heirs of Descartes’analytical and synthetic tools are still in balance ( differential universality theo-rem ), being the behavior of TMMs exactly the space of solutions definable withdifferential algebra (restricted to ordinary differential equations). Furthermore,to study the properties of the machines there is no need of geometric or ana-lytic insights, because the algebraic part provides algorithms to compute themautonomously (automated reasoning).In this paper, we have been able to define the behaviors of TMMs that havebeen introduced to formalize tractional constructions in a modern way. To myknowledge, it is the first clear definition of the limits of tractional motion. Suchlimits permit a distinction between objects that are constructible with TMMsand others that are not. To define the behavior of such machines, we used man-ifolds of functions: if Descartes’ setting defined a dualism between algebraic andtranscendental curves, our setting facilitates a new dualism between functions.As introduced in section 5.2, the obtainable functions are the differential alge-braic ones (shortly: D.A.), i.e. solutions of algebraic differential equations .As already mentioned, all elementary functions are D.A., and even most of thetranscendental functions that we find in most of the analysis handbooks. His-torically, the first example of non-D.A. function was the Γ of Euler, as provenin 1886 by H¨older. According to Rubel [1989], not-D.A. functions are named transcendentally transcendental . About functions in one variable, we can pro-vide a finer distinction beyond the algebraic and transcendental dualism. Ofcourse algebraic functions are also D.A., so, calling algebraic-transcendental thefunctions that are D.A. but not algebraic, we can divide functions in the casesof Table 1 (with some examples). About future extension beyond TMMs, a crucial role could be played by Euler Γfunction. As we are going to examine, this function can play for D.A. functionsthe same role as the exponential curve played for algebraic curves.Algebraic curves are defined as the zero set of polynomials, where a polyno-mial is an expression that involves only the operations of addition, subtraction,multiplication, and non-negative integer exponents. One can ask to relax theconstraint of considering only non-negative integer exponents (e.g. we may be Note that, even though without the geometrical interpretation, such dualism was in-troduced in analog computation by Shannon’s General Purpose Analog Computer Shannon[1941] many centuries after the introduction of tractional constructions. As visible in thetitle of Shannon’s paper, the GPAC was a theoretical model for the analog computer called differential analyzer .Furthermore, we also need a note on an implicit assumption. We considered function locallysmooth (i.e. of class C ∞ in a certain domain). In general, to be a solution of an algebraicdifferential equation of order n , we have to assume that the function is C n , not necessary C ∞ . For not-smooth functions, most of the results of differential algebra fail. These cases aretreated in Rubel [1983]. able 1: Categorization of functions in one variable (taken from [Shannon, 1941, p. 501]). Transcendental Algebraic Trascendentallytrascendental Algebraic-trascendental Euler Γ, Rie-mann ζ e x , log( x );trigonometric,hyperbolic andinverses; Bessel,elliptic and prob-ability functions Irrational alge-braic Rational x m ( m a ratio-nal fraction); so-lutions of alge-braic equationsin terms of a pa-rameter polynomials,quotientsof polyno-mialsinterested in considering monomials in x − , or in x √ ): from this perspec-tive, the exponential curve solves the problem of generic exponent. Concerningconstructions, tractional motion justified the exponential curve with the intro-duction of loads subject to friction or with blades or wheels. Therefore, eventhough the extension from polynomials to formulas with any exponent is notenough to define analytically all the functions constructible by tractional mo-tion, the construction of the exponential was important to focus on the role ofthe wheel for the expansion into the synthetic aspect.D.A. functions are solutions of differential polynomials. Differential polyno-mials are polynomials in the variables and their derivatives, but these derivativeshave to be of non-negative integer order. Negative integer-order derivatives canbe considered integrals. However, what does it mean to consider derivatives ofnon-integer order? This question is older than three centuries and is at the coreof fractional calculus .The idea of extending the meaning of d n ydx n to n / ∈ N appeared the firsttime in a letter of Leibniz to L’Hˆopital (September 30, 1695), and later gotmany mathematicians interested in it: Euler, Fourier, Abel, Liouville, Riemann,Laurent, Hadamard, Schwartz (for more precise historical references see Ross[1975], Ross [1977]). Nowadays, fractional calculus finds use in many fields ofscience and engineering, including fluid flow, rheology, diffusive transport akinto diffusion, electrical networks, electromagnetic theory, and probability. Thereis not a unique definition of fractional integral, but the following (usually called Riemann-Liouville fractional integral ) is probably the most used version. Weare proposing it just to give a first shallow idea, for clarifications and furtherreading, see Miller and Ross [1993].Starting from the Cauchy formula for repeated integration (it allows to com-press n antidifferentiations of a function into a single integral) D − n f ( x ) = 1( n − (cid:90) xa ( x − t ) n − f ( t ) dt, we can generalize n to non-integer values and, since n ! = Γ( n + 1), we get D − v f ( x ) = 1Γ( v ) (cid:90) xa ( x − t ) v − f ( t ) dt. This formula links Γ function and fractional calculus. The construction of Γ withidealized machines could be particularly important because a widely acceptedgeometric interpretation of fractional calculus is still missing (for some attempts,38ee Adda [1997], Podlubny [2002], Tavassoli et al. [2013], Herrmann [2014]).Hence, from a historical/philosophical perspective, fractional calculus is nowlooking for a constructive-synthetic geometrical legitimation, as it happenedin early modern period with transcendental curves. We hope that TMMs canconstitute a solid step over which such extensions may come. Since Newton and Leibniz, the core concept of calculus is the constructive role ofmethods involving the infinity. On the contrary, the proposed mechanical settingand the differential algebra counterpart suggest that it is possible to considercalculus (at least the part dealing with differential polynomials) without theneed of infinity, but with the less abstract idea that “the wheel direction isthe tangent.” A pedagogical peculiarity of such an approach to infinitesimalanalysis with mathematical machines is that students can manipulate conceptsusually considered too abstract. Some very preliminary attempts to introducetractional machines in math education have been proposed in a workshop Miliciand Di Paola [2012] and in some papers Milici [2012b], Di Paola and Milici[2012], Salvi and Milici [2013]. A proposal for science museum activities can bealso found in Milici and Dawson [2012].Furthermore, the possibility of a restructuration of infinitesimal analysisin the light of TMMs and differential algebra should be interesting to be in-vestigated from computational, instrumental, visual, algebraic, cognitive, andfoundational viewpoints Milici [2017].Finally, we want to conclude with a remark on the mutual evolution of ana-log and digital/symbolic computation in mathematics. An approach to breakthe Church-Turing thesis is to check whether some results beyond Turing com-putational limits may be reached somehow (the hypercomputation problem, e.g.Copeland [2002]). With regard to this question, it could be interesting to setthe problem from a purely mathematical point of view. Instead of consid-ering the physical limits of analog computing, one could look for an “exact”approach to analog computation through geometry because of its cognitive sim-plicity and richness. From this point of view, considering diagrammatic con-structions and symbolic manipulations respectively as analog and digital com-putations, the evolution of mathematical foundational paradigms from the geo-metric/arithmetic perspectives (with their relative intercourses and extensions)can be considered an evolution of computational limits.Considering the computational power of mathematical approaches, Pythagoreanrational numbers (arithmetic perspective) were not sufficient to express somevalues generated by the arithmetic reinterpretation of ruler-and-compass geo-metric constructions (the length of the diagonal of a square with respect to theedge). On the contrary, later polynomial algebra introduced values not geomet-rically constructible by ruler and compass (the exactness problem in the earlymodern period). However, the unbalance between the powers of the differentparadigms is not a constant. Descartes balanced their powers in analytical ge-ometry, and this powerful paradigm became the hard core over which calculusevolved, generating a rich symbolism inspired by ideas derived from geome-try and mechanics. Something new happened with regard to calculus: if thegeometrical paradigm had already been abandoned in other periods, for the39rst time there was the acceptance of entities generated by infinite processes(note that infinite procedures were also adopted by Archimedes, but only as aninvestigative tool to be later interpreted from a synthetic perspective). Thisacceptance of infinite processes made it difficult to re-interpret the obtained en-tities in everyday (finite) experience. 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