EEquilibria of static systems W lodzimierz M. Tulczyjew Valle San Benedetto, 262036 Monte Cavallo, Italy [email protected]
Abstract.
A conceptual framework for variational formulations of physical theories is proposed. Sucha framework is displayed here just for statics, but it is designed to be subsequently adapted tovariational formulations of static field theories and dynamics.
1. Introduction.
A conceptual framework for variational formulations of physical theories is proposed. In this paper,such a framework is displayed just for statics. It is nonetheless designed to be subsequently adapted tovariational formulations of static field theories and dynamics.The framework is crucially based on three fundamental notions, on which any theory of equilibriafor static systems should be founded. They are the concepts of (admissible) trajectory , work function and stable equilibrium , the latter coming with its variant of stable local equilibrium . For any admissibletrajectory of a system, a work function is assigned. This is a function which, at each point of the trajectory,gives the work that is needed to force the system to reach the configuration represented by the consideredpoint, evolving trough all of the preceding configurations in that trajectory. A configuration is a stableequilibrium for the system if for any trajectory stemming out from such configuration, the correspondingwork function has a minimum at the initial point. In order to study the points of equilibrium, the notions ofgerms and jets of the trajectories as well as of the functions on such trajectories are introduced. Differentialcriteria for equilibrium in terms of these concepts are given and various examples to illustrate such criteriaare presented.Each definition and each notation adopted in this paper is very carefully designed. The whole expo-sition is constantly guided by the following aims: (a) to subsume all previous definitions of equilibria forstatic systems; (b) to reach a theory of equilibria for static systems which is built upon purely geometricnotions. As mentioned above, this second aim is motivated also by the wish of paving the way towardsconstructions of purely geometric frameworks for theories of static fields and dynamics. Discussions ofthese theories will be objects of future work.We thank Franco Cardin and Andrea Spiro for conversations on the content of this paper and theirconstructive criticism.A similar approach to equilibria of static systems can be found in an article listed at the end of thecurrent paper.
2. Two simple examples.
Example 1.
Let Q be an affine space modelled on a vector space V with a Euclidean metric g W V ! V : .1/ A material point with configuration q Q is connected with a spring of spring constant k to a fixed point q Q . The configuration q D q is the only stable equilibrium configuration of the material point. N Example 2.
The material point with configuration q Q in Example 1 is subject to friction. Thefriction is measured by the coefficient . The set ˚ q Q I k q (cid:0) q k ı k (cid:9) .2/ is the set of equilibrium configurations. N efinitions of equilibrium:A) A stable equilibrium configuration is a configuration at which the internal energy of the systemassumes its minimum value.B) A configuration q is a stable equilibrium configuration if the work of each process starting at q and not ending at q is positive.Definition A) applies to the first example. The internal energy is the function U W Q ! R W q ‘ k2 k q (cid:0) q k : .3/ It assumes its minimum value at the configuration q D q . Definition A) does not apply to the secondexample.Definition B) applies to both examples. In the first example the work of a process starting at q andending at q equals U.q / (cid:0) U.q / . This work is always positive unless q D q . In the second examplethe work of a process from q to q equals U.q / (cid:0) U.q / C Œ length of process : .4/ Set q D q , q D q C q and assume that the process is the straight segment from q to q C q , then U.q / (cid:0) U.q / C Œ length of process D k2 k q (cid:0) q C q k (cid:0) k2 k q (cid:0) q k C k q kD k hhh g.q (cid:0) q /; q iii C k2 k q k C k q k .5/ Let k q (cid:0) q k > ı k: .6/ Choose q in the direction opposite to .q (cid:0) q / . We have U.q / (cid:0) U.q / C Œ length of process D (cid:0) k k q (cid:0) q k k q k C k q k C k2 k q k : .7/ This quantity is negative if k q k ¤ is small enough since (cid:0) k k q (cid:0) q k k q k C k q k < 0: .8/ It follows that q is not a configuration of equilibrium.Let k q (cid:0) q k ı k: .9/ If the process is the segment of a straight line from q to q C q and the non zero vector q points in thedirection of (cid:0) .q (cid:0) q / , then U.q / (cid:0) U.q / C Œ length of process D (cid:0) k k q (cid:0) q k k q k C k2 k q k C k q k > 0 .10/ In all other cases the value of the expression (4) is higher. It follows that q is a configuration of equilibrium.The two examples were designed to show that variational formulations have a wider area of applica-bility if based on Definition B). This definition appears in the Levi-Civita formulations of mechanics. Itis not present in modern geometric formulations. . Algebras and ideals. We will denote by K the set N [ f1 ; c g , where is the cardinality of N and c stands for thecardinality of R . The ordering relations , < , > , and > have in K the usual meaning of inequalities ofcardinal numbers.In the algebra A . R C / we introduce the sequence of ideals I . R C ; 0/ I . R C ; 0/ I . R C ; 0/ I c . R C ; 0/: .11/ The ideal I . R C ; 0/ D ˚ f A . R C / I f .0/ D (cid:9) .12/ is maximal in the sense that it is not a proper subset of any ideal except the trivial ideal A . R C / .For k N , the ideal I k . R C ; 0/ is the power . I . R C ; 0// k C of the ideal I . R C ; 0/ . The ideal I . R C ; 0/ is the intersection T k N I k . R C ; 0/ . The ideal I c . R C ; 0/ is the set of functions each vanish-ing in a closed neighbourhood of .A function h on R C is extended to a function on R and the derivative D l h is the derivative of theextended function. The derivative of order 0 of a function is the function itself. Proposition 1.
For k N a function h A . R C / is in I k . R C ; 0/ if and only if D i h.0/ D for i D
0; 1; : : : ; k . Proof:
The derivatives of a function h A . R C / at are well defined and the function can be repre-sented by the Taylor formula h D e .h/ C e .h/s C : : : C e k .h/s k C r s k C ; .13/ where e i .h/ D
1i Š D i h.0/; .14/s W R C ! R is the canonical injection, and r is a differentiable function on R C . The function s is in I . R C ; 0/ and the power s k C is in I k . R C ; 0/ . If D i h.0/ D for i D
0; 1; : : : ; k , then h D r s k C is in I k . R C ; 0/ .A function h I l . R C ; 0/ is a combination of products h h h l of elements of I . R C ; 0/ . Thederivative D h is a combination of products of functions with each product containing at least l factors in I . R C ; 0/ . It follows that the derivative D h of a function h I l . R C ; 0/ is in I l (cid:0) . R C ; 0/ . If h I k . R C ; 0/ ,thenD h D h I k . R C ; 0/; D h D D h I k (cid:0) . R C ; 0/; D h I k (cid:0) . R C ; 0/; : : : ; D k h I . R C ; 0/: .15/ Hence, D i h.0/ D for i D
0; 1; : : : ; k
4. Local equilibria.4.1. Trajectories. A physical system is a physical object with a selection of degrees of freedom intended to beinteracted with or controlled. The selected degrees of freedom are given a mathematical representationin the form of a differential manifold Q called the configuration space . The set of physical systemswith configuration space Q is denoted by S Q . In the configuration space Q we consider the set J .Q/ ofsubmanifolds c Q with boundary each homeomorphic to the set R C D Œ0; / R . We refer to thesesubmanifolds as displacement trajectories or simply trajectories of . Each trajectory c is the image ofan embedding q W R C ! Q .16/ alled a parameterisation . The configuration q D q .0/ is the initial configuration of c . The symbol J q .Q/ will denote the set of trajectories initiating at q . For each system S Q there is a set K J .Q/ of admissible trajectories . We refer to the set K as constraints .We expect the assignment of constraints to be local in the sense that if c and c are trajectories withthe same initial configuration, c is admissible, and c c , then c is admissible.We denote by K Q the set of origins of all admissible trajectories of . Constraints are said tobe holonomic if K is the set J .K / of all trajectories contained in K . There is an assignment W to each c K of a function W . c / I . c ; q/ A . c / .17/ called a work function . The configuration q is the origin of c . The ideal I . c ; q/ is the set f g A . c / I g.q/ D g : .18/ We expect the assignment of work functions to be local in the sense that if trajectories c and c withthe same initial configuration are admissible and c c , then the work function W . c / is the restrictionto c of the work function W . c / . A stable equilibrium configuration is a configuration of a physical system that can remain un-changed in a period of time without external interference. A stable equilibrium configuration q in adifferential manifold is said to be local if it is the only equilibrium configuration in a neighbourhood of q . Let q be a configuration of a system in the set K and let q be a local equilibrium configurationin each trajectory c K originating at q . Such configuration will be called a stable local equilibriumconfiguration of . We claim that a configuration q K is a stable local equilibrium configuration iffor every admissible trajectory c initiating at q the work function W . c / I . c ; q/ A . c / .19/ has a local minimum at q . The germ of trajectory c at its initial configuration q an equivalence class of trajectories. Two trajec-tories c and c are equivalent if they share the same initial configuration q and there is a neighbourhood U of q Q such that c \ U D c \ U . The germ of c at q is denoted by j c c .q/ . Germs of functions on a trajectory c at the initial configuration q c are equivalence classes offunctions on c . Functions g and g are equivalent if there is a neighbourhood I c of q such that g j I D g j I . The class of g is denoted by j c g.q/ . Germs of functions at q form an algebra A c . c ; q/ .The germ of a function g W c ! R can be identified with the germ of a function g W c ! R if thereis a neighbouhood U Q of q such that c \ U D c \ U and g j c \ U D g j c \ U . It follows that thealgebra A c . c ; q/ is associated with the germ j c c .q/ rather than with the trajectory c .We expect that the germ j c W . c /.q/ I c . c ; q/ D I . c ; q/ ı I c . c ; q/ A c . c ; q/ .20/ is the same for each trajectory belonging to the germ j c c .q/ K c . This natural locality condition meansthat the work of a displacement in the immediate neighbourhood of the initial configuration of a trajectory oes not depend on the continuation of the trajectory. This locality property permits the reformulation ofthe conceptual framework in terms of germs of trajectories. We will translate the original formulationsbased on trajectories in the language of germs and we will continue the analysis in this language. Severaladditional constructions are added. The original formulations are significant because germs are accessibleonly through their representatives.A system is going to be represented by a set K c J c Q of germs of admissible trajectories and agerm work function W c . j c c .q// D j c W . c /.q/ A c . c ; q/ .21/ for each germ j c c .q/ K c . The set of initial configurations of admissible trajectories is denoted by K .The constraints K c are said to be holonomic if K c D J c K . A configuration q K is a local stable equilibrium configuration of the system if W c . j c c .q// is positive for each germ j c c .q/ K c q D K c \ J c q Q . The germ is said to be positive if one of itsrepresentatives is positive in a neighbourhood of q with the exclusion of q . This representative has aminimum at q . We will write W c . j c c .q// > 0 to indicate that W c . j c c .q// is positive. Let k be an element of N . The k -jet j k c .q/ of a trajectory c at the initial configuration q is anequivalence class of trajectories. Trajectories c and c with the same initial configuration q are equivalentif I k .Q; q/ C I .Q; c / D I k .Q; q/ C I .Q; c /: .22/ The set of k -jets of trajectories will be denoted by J k Q and the set of k -jets of trajectories initiating at q will be denoted by J kq Q .The mapping k Q W J k Q ! Q W j k c .q/ ‘ q .23/ is the jet-source projection . Mappings k k Q W J k .Q/ ! J k Q W j k c .q/ ‘ j k c .q/ .24/ are well defined if k k . Jets of order k N of functions on a trajectory c at its initial configuration q are equivalence classesof functions. Functions g and g are equivalent if g (cid:0) g I k . c ; q/: .25/ The jet of g is denoted by j k g.q/ . The quotient algebra A k . c ; q/ D A . c / ı I k . c ; q/ .26/ is the algebra of jets at q .The algebra A . c / of functions on a trajectory c Q is isomorphic to the quotient algebra A.Q/ ı I .Q; c /: .27/ The algebra (26) is isomorphic to A .Q/ ı . I k .Q; q/ C I .Q; c // : .28/ he jet W k . c / D j k W . c / .29/ of a work function W . c / I . c ; q/ A . c / .30/ belongs to I k 0 . c ; q/ D I . c ; q/ ı I k . c ; q/: .31/ R C D Œ0; / R . Differentiable functions on R C D Œ0; / R are restrictions to R C of differentiable functions on R . The quotient A . R C / ı I k . R C ; 0/ is the algebra of k -jets of functions at R C . The jet of a function h W R C ! R is denoted by j k h.0/ .We consider local minima of functions in I . R C ; 0/ at 0. A function h I . R C ; 0/ is said to have a local minimum at if there is a number a > 0 such that h is increasing in .0; a/ R .The jet j k h.0/ I k0 . R C ; 0/ is fully represented by the sequence e .h/; : : : ; e k .h/ .32/ of derivatives e i .h/ D
1i Š D i h.0/ .33/ of its representative h I . R C ; 0/ . The jet is said to be positive if the first non zero element in thesequence is positive. The jet is said to be negative if the first non zero element in the sequence is negative.Each element of the ideal I k 0 . R C ; 0/ is either positive or negative if it is not zero. There are obviousrelations > , < , > , and between elements of I k 0 . R C ; 0/ . Proposition 2.
If a jet j k h.0/ I k 0 . R C ; 0/ is positive, then the function h has a local minimumat . Proof: If e i .h/ D for i D
1; : : : ; l (cid:0) , then the function h I . R C ; 0/ is represented by the Taylorformula h.s/ D e l .h/s l C r .s/s l C : .34/ From lim s ! (cid:0) .l C C D r .s/s D it follows that there is a number ı > 0 such that j .l C C D r .s/s j < j le l .h/ j .36/ for j s j < ı . If e l .h/ > 0 , then the function h is increasing in the interval Œ0; ı since the derivativeD h.s/ D le l s l (cid:0) C .l C l C D r .s/s l C D (cid:0) le l C .l C C D r .s/s s l (cid:0) .37/ is positive for . It follows that the function h has a local minimum at .The proposition establishes a sufficient condition for a local minimum of a function h I . R C ; 0/ at for each k N [ f1g . Proposition 3.
If a jet j k h.0/ I k0 . R C ; 0/ is negative, then the function h does not have alocal minimum at . esults of the present section are summarised in the following statements:A) If the jet j k h.0/ of a function h I . R C ; 0/ is positive for some k N , then the function h hasa local minimum at 0.B) If a function h I . R C ; 0/ has a local minimum at 0, then for each k N the jet j k h.0/ ispositive or zero. With a trajectory c with an initial configuration q we associate ideals I k . c ; q/ and quotient algebras A k . c ; q/ D A . c / ı I k . c ; q/ .38/ with k N .A jet j k c .q/ is represented by the ideal I k .Q; q/ C I .Q; c /: .39/ The algebra A . c / is canonically isomorphic to the quotient algebra A .Q/ ı I .Q; c / .40/ and the ideal I k . c ; q/ is isomorphic to . I k .Q; q/ C I .Q; c // ı I .Q; c /: .41/ In consequence of these isomorphisms the algebras A k . c ; q/ D A . c / ı I k . c ; q/ .42/ and A .Q/ ı . I k .Q; q/ C I .Q; c // .43/ are isomorphic. If j k c .q/ D j k c .q/ , then I k .Q; q/ C I .Q; c / D I k .Q; q/ C I .Q; c /; .44/ hence algebras A k . c ; q/ and A k . c ; q/ are isomorphic. It follows that the algebra A k . c ; q/ is associatedwith the jet j k c .q/ rather than with the trajectory c . The algebra A .Q; j k c .q// D A .Q/ ı . I k .Q; q/ C I .Q; c // .45/ is a convenient representant of the different, though isomorphic, algebras A k . c ; q/ associated with a jet j k c .q/ . We will apply the established isomorphisms to ideals I k0 . c ; q/ . The ideal I .Q; j k c .q// D I .Q; q/ ı . I k .Q; q/ C I .Q; c // .46/ will represent the class of isomorphic ideals.A function g I . c ; q/ has a local minimum at q if there is a neighbourhood u c of q such that g is increasing on u . We will establish conditions for minima in terms of jets of functions. Let c be aprocess with initial configuration q and let g be a function in the ideal I . c ; q/ A . c / . Let q W R C ! Q .47/ nd q W R C ! Q .48/ be parameterisations of the trajectory c , and let W R C ! R C .49/ be the reparameterisation diffeomorphism such that .0/ D and q D q B : .50/ The parameterisations induce the corestrictions c j q W R C ! c W s ‘ q .s/: .51/ and c j q W R C ! c W s ‘ q .s/: .52/ The compositions h D g ı c j q and h D g ı c j q are functions on R C . The equality h D h B .53/ follows from (50). There are Taylor series h D e .h/s C : : : C e k .h/s k C : : : ; .54/h D e .h /s C : : : C e k .h /s k C : : : ; .55/ and D e ./s C : : : C e k ./s k C : : : .56/ associated with the functions and the diffeomorphism. By comparing the expression h D e .h /s C : : : C e k .h /s k C : : : .57/ with h B D e .h/ C : : : C e k .h/ k C : : : .58/ we arrive at the following observations: If e i .h/ is the first non zero term in the sequence e .h/; : : : ; e k .h/; : : : ; .59/ then e i .h / is the first non zero term in the sequence e .h /; : : : ; e k .h /; : : : ; .60/ and e i .h / D e i .h/.e .// i : .61/ If e i .h/ is positive, then e i .h / is positive since e ./ is positive. It follows from these observations thatthe jet j k h .0/ is positive if and only if the jet j k h.0/ is positive. It is then correct to declare the jet j k g.q/ positive if the jet j k .g ı c j q /.0/ of the function g ı c j q constructed with a parameterisation q is positive.We will adopt the following notational conventions to jets of functions on trajectories. We will write j k g.q/ > 0 to indicate that the jet j k g.q/ is positive. We will write j k g.q/ > to indicate that the jet j k g.q/ is non negative. he corestriction c j q is a diffeomorphism. The function g W c ! R .62/ has a local minimum at q if and only if the function g B c j q W R C ! R .63/ has a local minimum at . Adaptations of conditions A) and B) of Subsection 4.6.3 follow.A) The function g has a local minimum at q if j k g.q/ > 0 for some k N .B) If a function g I . c ; q/ has a local minimum at q , then j k g.q/ > for each k N . The function W .c/ is defined on a trajectory c K . The jet j k W . c /.q/ is assigned to the germ j c c .q/ K c .64/ and not to the entire trajectory c . This is a consequence of the locality assumed in an earlier section.Equilibrium criteria of differential order k N are formulated in terms of the set K k of k -jets ofadmissible trajectories of the system and the work functions W k . j k c .q// . The following differentialequilibrium conditions of order k are based on propositions formulated in the introductory sections.A) A sufficient condition: a configuration q K is a stable local equilibrium configuration if W k . j k c .q// > 0 .65/ for each j k c .q/ K kq D K k \ J kq Q .This condition is inconclusive if W k . j k c .q// D for each j k c .q/ K kq D K k \ J kq Q . Higher order criteria have to be examined to establishstability.B) A necessary condition: if a configuration q K is a stable local equilibrium configuration ofthe system, then W k . j k c .q// > for each j k c .q/ K kq .
5. Parameterised trajectories.5.1. Parametrisations.
Trajectories are images of embeddings q W R C ! Q .68/ called parameterisations. The set of parameterisations of all trajectories is denoted by E .Q/ .The group G of reparameterisations is the set of diffeomorphisms from R C onto R C . There is theright action B W E .Q/ G ! E .Q/ W . q ; / ‘ q B : .69/ We introduce an equivalence relation in the set of parameterisations. Parameterisations q and q areequivalent if there is a reparameterisation G such that q D q B . Equivalence classes are identifiedwith trajectories. The trajectory corresponding to a class is the image of one if its elements. The set J .Q/ is the quotient set E .Q/= G . .2. Constraints. Constraints for a system S Q are a set L E .Q/ of admissible embeddings. The set L is theset of initial points of admissible embeddings. For each q L the set L q D f q L I q .0/ D q g .70/ is a G -cone: if q L q , then q B L q for each G .Constraints are holonomic if L is the set E .L / of all trajectories contained in L . A function X . q / I . R C ; 0/ D ˚ f A . R C / I f .0/ D (cid:9) .71/ called a work function is assigned to each embedding q L . The work function is required to behomogeneous: X . q B / D X . q / B .72/ for each G .If a work function W . c / W c ! R .73/ is specified for each admissible trajectory c K then X . q / W R C ! R W s ‘ W . q .s// .74/ is assigned to the parameterisation q of c . The inverse relation is expressed by W . c / W c ! R W q ‘ X . q (cid:0) .q//: .75/ A system S Q is characterised by the set L E .Q/ and the work function X . q / W R C ! R for each q L .A configuration q L is a local stable equilibrium configuration if X . q / has a local minimumat . Germs of embeddings are equivalence classes. Embeddings q and q are equivalent if there is aneighbourhood V of R C such that q j V D q j V . The germs of q at q is denoted by t c q .0/ .The set of germs of embeddings is denoted by T c Q and the set of germs of admissible embeddingsis denoted by L c . For each q L the set L c q is a G -cone: if t c q .0/ is in L c q , then t c . q B / is in L c q for each G . Germs of functions on R C at are equivalence classes of functions. Functions h and h are equivalentif there is a neighbourhood K of in R C such that h j K D h j K . The class of h is denoted by j c h.0/ .Germs of functions at form an algebra A c . R C ; 0/ .There is the work function X c . t c q .0// D t c X . q /.0/ I c . R ; 0/ A c . R ; 0/ .76/ or each germ t c q .0/ L c . The work function is homogeneous: X c . t c . q B /.0// D X c . t c q .0// B t c .0/ .77/ for each G . A configuration q L is a local stable equilibrium configuration if X c . g / is positive for eachgerm g L c q . Vectors are equivalence classes of embeddings. Embeddings q and q are equivalent if f ı q (cid:0) f ı q I k . R C ; 0/ .78/ for each function f W Q ! R . The k -vector of an embeddng q will be denoted by t k q .0/ . The space of k -vectors will be denoted by z T k Q .There are the k - tangent fibrations z T k QQ ................................................................................................... z k Q .79/ with projections z k Q W z T k Q ! Q W t k .0/ ! .0/ .80/ and the projections z k k Q W z T k Q ! z T k Q W t k .0/ ! t k .0/ .81/ for k k . Relations z k Q ı z k k Q D z k Q .82/ and z k k Q ı z k k Q D z k kQ .83/ are satisfied for k k k .There is a distinguished section of z k Q defined as O kQ W Q ! z T k Q W q ‘ t k ! q .0/ .84/ with ! q W R ! Q W s ! q: .85/ There is a right action of the group G on fibres of z k Q defined by ı W z T k Q G ! z T k Q W . t k .0/; / ‘ t k . ı /.0/: .86/ This action leaves the distiguished section invariant. roposition 1. in Section 3. offers an alternate definition of k -tangent vectors as equivalence classesof embeddings. Embeddings q and q are equivalent ifD l .f ı q /.0/ D D l .f ı q /.0/ .87/ for each differentiable function f W Q ! R and each l k .It follows from the alternative definition that if z . t k q .0// D t q .0/ D
0; .88/ then D .f B q /.0/ D for each differentiable function f W Q ! R . This is not true if q is an embedding. Hence, z . t k q .0// ¤
0: .90/
Equilibrium criteria of differential order k N are formulated in terms of the set L k of k -vectors ofadmissible embeddings and the work functions X k . t k q .0// D j k X . q /.0/ I k0 . R ; 0/ .91/ for each vector t k q .0/ L k . These objects have homogeneity properties similar to those described forgerms.Differential equilibrium conditions of order k follow.A) A sufficient condition: a configuration q L is a stable local equilibrium configuration if X k . t k q .0// > 0 .92/ for each t k q .0/ L kq .B) A necessary condition: if a configuration q L is a stable local equilibrium configuration ofthe system, then X k . t k q .0// > for each t k q .0/ L kq D z T kq Q \ L k .
6. Examples of first order criteria.6.1. Affine configuration spaces.
In most examples the configuration space will be an affine space Q with the model space V equippedwith a Euclidean metric tensor g W V ! V : .94/ The space z T Q of variations is represented by Q z V .95/ with z V D f ıq V I ıq ¤ g : .96/ The tangent projection is the canonical projection z Q W Q z V ! Q W .q; ıq/ ‘ q: .97/ he bundle T Q is identified with Q V : .98/ The mapping Q W Q V W .q; a/ ‘ q .99/ is the canonical projection.The fibre product T Q . Q ; z Q / z T Q D n .; ıq/ T Q z T Q I Q ./ D z Q .ıq/ o .100/ is represented by Q V z V .101/ and hhh ; iii W Q V z V ! R W .q; ; ıq/ ‘ .ıq/ .102/ is the canonical pairing.The set G D f r R I r > 0 g .103/ is a group with group operation W G G ! G W .r; r / ‘ r r ; .104/ the unit , and the right action W .Q V / G ! Q V W ..q; ıq/; r / ‘ .q; ıq r /: .105/ The canonical pairing is G homogeneous: hhh .q; f /; .q; ıq/ r iii Q D hhh .q; f /; .q; ı/ iii Q r: .106/ Example 3.
A material point with configuration q in Q is tied to a fixed point q with a spring of springconstant k . This is an unconstrained potential system. The work function X . q / W R C ! R W s ‘ U. q .s// (cid:0) U. q .0// .107/ defined for each q L D E .Q/ is derived from the potential U W Q ! R W q ‘ k2 k q (cid:0) q k : .108/ The first order work function X W Q z V ! R W .q; ıq/ ‘ k hhh g.q (cid:0) q /; ıq iii .109/ follows.The first order necessary condition X .q; ıq/ D k hhh g.q (cid:0) q /; ıq iii > or each vector ıq V is satisfied with q D q .The first order sufficient condition X .q; ıq/ D k hhh g.q (cid:0) q /; ıq iii > 0 .111/ is inconclusive. N Example 4.
Let a material point be subject to isotropic static friction represented by a positive function W Q ! R : .112/ There are no constraints. The work function X . q / W R C ! R W s ‘ Z s0 . B q / k t q k .113/ is defined for each q L D E .Q/ . The first order work function X W Q V ! R W .q; ıq/ ‘ .q/ k ıq k D .q/ p hhh g.ıq/; ıq iii .114/ follows.The first order equilibrium condition .q/ k ıq k > 0 .115/ with ıq ¤ is a sufficient condition. All configurations are stable equilibrium configurations. N Example 5.
Let a material point with configuration q be tied with a rigid rod of length a to a point withconfiguration q . This is a static spherical pendulum with bilateral holonomic constraint L D f q Q I k q (cid:0) q k D a g .116/ and the work function X . q / W R C ! R W s ‘ U. q .s// (cid:0) U. q .0// .117/ derived from the potential U W L ! R W q ‘ m hhh g.v/; q (cid:0) q iii .118/ The unit vector v V is pointing up. The covector (cid:0) mg.v/ is the constant internal force due to gravity.The first order constraint is the set L D z T L D n .q; ıq/ Q z V I k q (cid:0) q k D a; hhh g.q (cid:0) q /; ıq iii D o .119/ and X W L ! R W .q; ıq/ ‘ m hhh g.v/; ıq iii .120/ is the work function.Configurations q D q C av and q D q (cid:0) av are in L and satisfy the first order necessary condition X .q; ıq/ D m hhh g.v/; ıq iii > for each vector .q; ıq/ L . N Example 6.
A point with configuration q is tied with a flexible string of length a to a point withconfiguration q . The configurations are constrained to the closed ball L D ˚ q Q I k q (cid:0) q k a (cid:9) : .122/ his is a system with holonomic unilateral constraints. Internal forces are derived from the potential U W L ! R W q ! k2 k q (cid:0) q k : .123/ First order constraint is the set L D z T L D n .q; ıq/ Q z V I k q (cid:0) q k a; hhh g.q (cid:0) q /; ıq iii if k q (cid:0) q k D a o : .124/ The work function X W L ! R W .q; ıq/ ‘ k hhh g.q (cid:0) q /; ıq iii .125/ is derived from the potential (123).A configuration q L D ˚ q Q I k q (cid:0) q k a (cid:9) .126/ saisfies the first order necessary condition X .q; ıq/ D k hhh g.q (cid:0) q /; ıq iii > for each ıq V such that hhh g.q (cid:0) q /; ıq iii if k q (cid:0) q k D a: .128/ When k q (cid:0) q k < a , then ıq is arbitrary. If k q (cid:0) q k < a , then the condition (127) is satisfied at q D q . This condition can not be satisfied if k q (cid:0) q k > a .When k q (cid:0) q k D a , then hhh g.q (cid:0) q /; ıq iii
0: .129/
The necessary condition implies that the vector q (cid:0) q is parallel to q (cid:0) q in the opposite direction. Hence, q (cid:0) q D (cid:0) k q (cid:0) q k a (cid:0) .q (cid:0) q /: .130/ This leads to q (cid:0) q D (cid:0) C a (cid:0) k q (cid:0) q k .q (cid:0) q /: .131/ The configuration q is at a distance k q (cid:0) q k D a C k q (cid:0) q k > a .132/ from q , and the configuration q is placed on the line segment between q and q . N Example 7.
Let M be an affine plane modelled on a Euclidean vector space V . The configuration spaceof a skiboard is the set Q D M P , where P is the projective space of directions in the affine space M .We use the Euclidean metric in M to identify the space P with the unit circle P D f V I hhh g. iii D g : .133/ Tangent vectors are elements of the space T Q D M V T P; .134/ where T P D f . P V I hhh g. iii D g : .135/ The set z T P D f . T P I k ıx k C k ı k ¤ g .136/ s the set of tangent vectors of embeddings.The skiboard is not constrained but is subject to anisotropic friction represented by positive functions q , x , and on M . The work function X .; / W R C ! R W s ‘ Z s0 . q B / jhhh g B ; t iiij C Z s0 . x B / q k t k (cid:0) hhh g B ; t iii C Z s0 . B / k t k .137/ is defined for each .; / L D E .M P / . The mapping X W M V T P ! R W .x; ıx; ‘ q .x/ jhhh g. iiij C x .x/ q k ıx k (cid:0) hhh g. iii C .x/ k ı k .138/ is the first order work function. The first order equilibrium condition X .x; ıx; D q .x/ jhhh g. iiij C x .x/ q k ıx k (cid:0) hhh g. iii C .x/ k ı k > 0 .139/ is sufficient. All configurations are stable equilibrium configurations. N Example 8.
Let M be an affine plane modelled on a Euclidean vector space V . The configuration spaceof a skate is the set Q D M P , where P is the projective space of oriented directions in the affine space M . We use the Euclidean metric in M to identify the space P with the unit circle: P D f V I hhh g. iii D g : .140/ This is the configuration space of the skiboard of the preceding example.The set D D f . t .0/; t .0// T M T P I t .0/ D ˙ k t .0/ k .0/ gD ˚ . t .0/; t .0// T M T P I ∃ R t .0/ D .0/ (cid:9) .141/ of tangent vectors in the configuration space of the skate is an integrable homogeneous differential equa-tion. A solution q D .; / W R C ! M P .142/ is constructed from an arbitrary parameterisation W R C ! P .143/ and the mapping W R C ! M W t ‘ x C Z t .144/ with an arbitrary choice of a function W R C ! R . Solutions of the equation constitute the constraints L of the skate.Tangent vectors are elements of the space M V T P; .145/ where T P D f . P V I hhh g. iii D g : .146/ The cotangent bundle T Q is the space Q V T P with the space T P constructed as thequotient T P D .P V / ı T B P .147/ f P V and T B P D n . t / P V I ∀ . T P hhh t ; ı iii D o .148/ fibred over P . This is the natural choice of the cotangent fibration since T P is a vector subfibration of P V . If T P is the class of . t / P V , then hhh ; . iii D hhh t ; ı iii : .149/ The first order constraint is the set L D ˚ .x; ıx; M V T P I ∃ k R ıx D k (cid:9) .150/ First order necessary conditions are satisfied for all configurations.Let the skate be subject to friction represented by a non negative function W M ! R and let it becontrolled by an external force f V . No external torque is applied. The work is the function X W L ! R W .x; ıx; ‘ .x/ k ıx k D .x/ p hhh g.ıx/; ıx iii : .151/ The inequality .x/ k ıx k C hhh f; ıx iii > for arbitrary .x; ıx; L .153/ is the necessary condition for equilibrium at .x; L with an external force f .By using ıx D k we arrive at the inequality .x/ j k j C k hhh f; iii > for each k R . The inequality must be satisfied for k D (cid:0) hhh f; iii . Hence, .x/ jhhh f; iiij (cid:0) hhh f; iii > and .x/ (cid:0) jhhh f; iiij >
0: .156/ If .x/ > jhhh f; iiij , then .x/ j k j > j k j jhhh f; iiij > hhh f; k iii .157/ for each k R . It follows that the inequality (152) is satisfied.We have obtained the inequality .x/ > jhhh f; iiij .158/ as an explicit condition of equilibrium at .x; L with the external force f .Examining the sufficient condition .x/ k ıx k C hhh f; ıx iii > 0 .159/ with ıx ¤ , we find that the skate is in stable equilibrium if .x/ > jhhh f; iiij .160/ and is not in stable equilibrium if .x/ D jhhh f; iiij : .161/ Example 9.
Let Q be the affine physical space. The model space is a Euclidean vector space V with ametric tensor g W V ! V : .162/ The example gives a formal description of experiments performed by Coulomb in his study of staticfriction.A material object is constrained to the set L D f q Q I hhh g.k/; q (cid:0) q iii > g ; .163/ where q is a point in Q and k is a unit vector in the model space V . The boundary @L D f q Q I hhh g.k/; q (cid:0) q iii D g .164/ is a plane passing through q and orthogonal to k . In its displacements along the boundary the pointencounters friction proportional to the component of the internal force pressing the point against theboundary. The friction coefficient > 0 defines a cone C D ıq V I hhh g.k/; ıq iii > q k ıq k (cid:0) hhh g.k/; ıq iii : .165/ The set of parameterised trajectories L D n q T .Q/ I ∀ s R C q .s/ L ; t q .s/ C if q .s/ @L o : .166/ is a non holonomic constraint. The trajectories are solutions of the homogeneous integrable differentialequation D D n .q; ıq/ Q z V I q L ; if q @L ; then ıq C o : .167/ The work is the function X . q / D for each q L .The first order constraint is the set L D n .q; ıq/ Q z V I q L ; if q @L ; then ıq C o D .q; ıq/ Q z V I q L ; if q @L ; then hhh g.k/; ıq iii > q k ıq k (cid:0) hhh g.k/; ıq iii .168/ and X D is the work function without internal forces.Necessary equilibrium conditions are satisfied at all configurations q L with zero internal forces.The inequality hhh f; ıq iii > for arbitrary .q; ıq/ L .171/ is the necessary condition for equilibrium at q L with an internal force f .If the object is not on the boundary, then hhh g.k/; q (cid:0) q iii > 0 . The virtual displacements are notconstrained and .q; f / Q V satisfies the necessary condition of equilibrium if and only if f D . f the object is on the boundary, then we show that .q; f / satisfies the necessary condition of equi-librium if and only if the inequality q k f k (cid:0) hhh f; k iii (cid:0) hhh f; k iii is satisfied.If f D (cid:0) k f k g.k/ , then .q; f / satisfies the necessary condition of equilibrium and (172) is satisfied.We will examine the case f ¤ (cid:0) k f k g.k/ . Let .q; f / satisfy the necessary condition of equilibrium.The virtual displacement .q; ıq/ with ıq D (cid:0) g (cid:0) .f / C hhh f; k iii k C q k f k (cid:0) hhh f; k iii k .173/ is in L since hhh g.k/; ıq iii D q k f k (cid:0) hhh f; k iii D q k ıq k (cid:0) hhh g.k/; ıq iii : .174/ From the principle of virtual work (170) and hhh f; ıq iii D (cid:0) k f k C hhh f; k iii C q k f k (cid:0) hhh f; k iii hhh f; k iii .175/ it follows that (cid:0) k f k C hhh f; k iii C q k f k (cid:0) hhh f; k iii hhh f; k iii > and q k f k (cid:0) hhh f; k iii (cid:0) hhh f; k iii D q k f k (cid:0) hhh f; k iii (cid:0) k f k (cid:0) hhh f; k iii (cid:0) q k f k (cid:0) hhh f; k iii hhh f; k iii We have obtained the inequality (172).The Schwarz inequality jhhh g.u/; v iii (cid:0) hhh g.k/; u iiihhh g.k/; v iiij q k u k (cid:0) hhh g.k/; u iii q k v k (cid:0) hhh g.k/; v iii .178/ for the bilinear symmetric form .u; v/ ‘ hhh g.u (cid:0) hhh g.k/; u iii k/; v (cid:0) hhh g.k/; v iii k iii D hhh g.u/; v iii (cid:0) hhh g.k/; u iiihhh g.k/; v iii .179/ applied to the pair . (cid:0) g (cid:0) .f /; ıq/ results in the inequality (cid:0) hhh f; ıq iii C hhh f; k iiihhh g.k/; ıq iii q k f k (cid:0) hhh f; k iii q k ıq k (cid:0) hhh g.k/; ıq iii : .180/ If q k f k (cid:0) hhh f; k iii hhh f; k iii ; .181/ and hhh g.k/; ıq iii > q k ıq k (cid:0) hhh g.k/; ıq iii ; .182/ then (cid:0) hhh f; ıq iii C hhh f; k iiihhh g.k/; ıq iii hhh f; k iii q k ıq k (cid:0) hhh g.k/; ıq iii hhh f; k iiihhh g.k/; ıq iii : .183/ ence, hhh f; ıq iii > and .q; f / satisfies the necessary condition of equilibrium.The inequality q k f k (cid:0) hhh f; k iii (cid:0) hhh f; k iii means that the covector f is inside a cone in the space V . The covector g.k/ is the axis of the cone andthe angle cot (cid:0) ./ is the aperture. N Example 10.
The present example gives a simplified discrete model of the buckling of a rod. One endof the rod is a point in an affine space Q with configuration q positioned on the half-line L D f q Q I q (cid:0) q D hhh g.u/; q (cid:0) q iii u; hhh g.u/; q (cid:0) q iii > 0 g .186/ starting at a point q in the direction of a unit vector u . The other end is a point with configuration q constrained to the plane L D f q Q I hhh g.u/; q (cid:0) q iii D g .187/ through q perpendicular to u . The rod can be compressed or extended in length but not bent. Its relaxedlength is a and the elastic constant is k . The buckling of the rod is simulated by displacements of its endpoint in the plane L tied elastically to the point q with a spring of spring constant k . The configurationspace is the affine space Q with holonomic constraints represented by L . The set L D n .q; ıq/ Q z V I q L ; hhh g.u/; ıq iii D o .188/ of admissible virtual displacements is the tangent set of L .The internal energy of the system is the function U W L ! R W q ‘ k2 . k q (cid:0) q k (cid:0) a/ C k k q (cid:0) q k : .189/ The first order work function X W L ! R W .q; ıq/ ‘ (cid:0) k.1 (cid:0) a k q (cid:0) q k (cid:0) / C k hhh g.q (cid:0) q /; ıq iii .190/ is derived from the internal energy. The equality hhh g.q (cid:0) q /; ıq iii D hhh g.q (cid:0) q /; ıq iii .191/ is used. The first order necessary condition X .q; ıq/ D (cid:0) k.1 (cid:0) a k q (cid:0) q k (cid:0) / C k hhh g.q (cid:0) q /; ıq iii D for all .q; ıq/ in L is satisfied if q D q or k.1 (cid:0) a k q (cid:0) q k (cid:0) / C k D
0: .193/ hence k q (cid:0) q k D akk C k : .194/ he equality k q (cid:0) q k D akk C k (cid:0) k q (cid:0) q k .195/ is derived from k q (cid:0) q k C k q (cid:0) q k D k q (cid:0) q k D akk C k : .196/ Only q D q is possible if k q (cid:0) q k > akk C k : .197/ If k q (cid:0) q k akk C k .198/ the configuration q is found in the set S D ( q P I k q (cid:0) q k D akk C k (cid:0) k q (cid:0) q k ) : .199/ The first order sufficient condition is inconclusive. N
7. Second order criteria.
Second order equilibrium criteria will be formulated in affine configuration spaces. A Euclideanmetric tensor g W V ! V .200/ in the model space V of an affine space Q will be present.The space z T Q of second variations is represented by Q z V V . The tangent projection is thecanonical projection z W Q z V V ! Q W .q; ı q; ı q/ ‘ q: .201/ The set G D ˚ .r ; r / R I r > 0 (cid:9) .202/ is a group with group operation W G G ! G W ..r ; r /; .r ; r // ‘ .r r ; r r C r r /; .203/ the unit .1; 0/ , and the right action W .Q z V / G ! Q z V W ..q; ı q; ı q/; .r ; r // ‘ .q; ı q r ; ı q r C ı q r / .204/ with z V D z V V .
8. Equilibrium criteria.
A constraint is a set L Q V V: .205/
For each q in L D .L / .206/ the set L D L \ T Q .207/ s a cone: .q; ı q; ı q/ L .208/ implies .q; ı q; ı q/ .r ; r / L .209/ for each .r ; r / G .The work function X W L ! I . R C ; 0/ D R .210/ is homogeneous: X ..q; ı q; ı q/ .r ; r // D X .q; ı q; ı q/ .r ; r / .211/ for all .q; ı q; ı q/ L and .r ; r / G .A) The necessary equilibrium condition: if a configuration q L is a stable local equilibrium con-figuration of the system, then X .q; ı q; ı q/ > .0; 0/ .212/ for each .q; ı q; ı q/ L B) The sufficient condition: a configuration q L is a stable local equilibrium configuration if X .q; ı q; ı q/ > .0; 0/ .213/ for each .q; ı q; ı q/ L .
9. Examples.
Example 11.
Let a material point with configuration q be tied with a rigid rod of length a to a pointwith configuration q . This is a static spherical pendulum with bilateral holonomic constraint L D f q Q I k q (cid:0) q k D a g .214/ and the work function X . q / W R C ! R W s ‘ U. q .s// (cid:0) U. q .0// .215/ derived from the potential U W L ! R W q ‘ m hhh g.v/; q (cid:0) q iii .216/ The unit vector v V is pointing up. The covector (cid:0) mg.v/ is the constant internal force due to gravity.The second order constraint is the set L D T z T L D n .q; ı q; ı q/ Q z V V I k q (cid:0) q k D a hhh g.q (cid:0) q /; ı q iii D k ı q k C hhh g.q (cid:0) q /; ı q iii D (cid:9) : .217/ It is used with the work function X W L ! R W .q; ı q; ı q/ ‘ (cid:0) m hhh g.v/; ı q iii ; m hhh g.v/; ı q iii : .218/ The necessary equilibrium condition X .q; ı q; ı q/ D (cid:0) m hhh g.v/; ı q iii ; m hhh g.v/; ı q iii > .0; 0/ .219/ for each .ı q; ı q/ L is satisfied at q D q (cid:0) av L since m hhh g.v/; ı q iii D or each ı q in L D n ı q V I hhh g.q (cid:0) q /; ı q iii D o .221/ and m hhh g.v/; ı q iii > 0 .222/ for each .ı q; ı q/ L . This is also a sufficient condition. The configuration q D q (cid:0) av is a stablelocal equilibrium configuration.The equality m hhh g.v/; ı q iii D for each ı q in L holds also at q D q C av . The inequality m hhh g.v/; ı q iii < 0 .224/ for each .ı q; ı q/ L implies that q D q C av is not a stable equilibrium configuration. N Example 12.
The present example gives a simplified discrete model of the buckling of a rod describedin Example 10. The set L D n .q; ı q; ı q/ Q z V V I q L ; hhh g.u/; ı q iii D hhh g.u/; ı q iii D o .225/ is the second tangent set of L .The internal energy of the system is the function U W L ! R W q ‘ k2 . k q (cid:0) q k (cid:0) a/ C k k q (cid:0) q k : .226/ The second order work function is the mapping X W L ! R W .q; ı q; ı q/ ‘ ..k.1 (cid:0) a k q (cid:0) q k (cid:0) / C k / hhh g.q (cid:0) q /; ı q iii ;ka k q (cid:0) q k (cid:0) hhh g.q (cid:0) q /; ı q iii C .k.1 (cid:0) a k q (cid:0) q k (cid:0) / C k /. k ı q k C hhh g.q (cid:0) q /; ı q iii //: .227/ It is known from Example 10 that necesary conditions are satisfied when q D q or k q (cid:0) q k D akk C k : .228/ The second order sufficient condition X .q; ı q; ı q/ D ..k.1 (cid:0) a k q (cid:0) q k (cid:0) / C k / hhh g.q (cid:0) q /; ı q iii ;ka k q (cid:0) q k (cid:0) hhh g.q (cid:0) q /; ı q iii C .k.1 (cid:0) a k q (cid:0) q k (cid:0) / C k /. k ı q k C hhh g.q (cid:0) q /; ı q iii / > 0 .229/ will be applied to these configurations.At q D q X .q; ı q; ı q/ D .0; .k.1 (cid:0) a k q (cid:0) q k (cid:0) / C k / k ı q k / > 0: .230/ f k q (cid:0) q k > akk C k ; .231/ then k.1 (cid:0) a k q (cid:0) q k (cid:0) / C k > 0 .232/ and the system is at stable equilibrium at q D q . If k q (cid:0) q k < akk C k ; .233/ then k.1 (cid:0) a k q (cid:0) q k (cid:0) / C k < 0 .234/ and q D q is not an equilibrium configuration. If k q (cid:0) q k D akk C k ; .235/ then k.1 (cid:0) a k q (cid:0) q k (cid:0) / C k D and the stability test is inconclusive.If q ¤ q and k q (cid:0) q k D akk C k ; .237/ then k.1 (cid:0) a k q (cid:0) q k (cid:0) / C k D and the sufficient condition X .q; ı q; ı q/ D .0; ka k q (cid:0) q k (cid:0) hhh g.q (cid:0) q /; ı q iii > 0: .239/ is satisfied. N
10. Third order criteria.
Example 13.
We are returning to the simplified discrete model of the buckling of a rod described in thepreceding example.The set L D n .q; ı q; ı q; ı q/ Q z V V V I q L ; hhh g.u/; ı q iii D hhh g.u/; ı q iii D hhh g.u/; ı q iii D o .240/ is the third tangent set of L D f q Q I hhh g.u/; q (cid:0) q iii D g : .241/ The internal energy of the system is the function U W L ! R W q ‘ k2 . k q (cid:0) q k (cid:0) a/ C k k q (cid:0) q k .242/ bd the third order work function is the function X W L ! R W .q; ı q; ı q; ı q/ ‘ ..k.1 (cid:0) a k q (cid:0) q k (cid:0) / C k / hhh g.q (cid:0) q /; ı q iii ;ka k q (cid:0) q k (cid:0) hhh g.q (cid:0) q /; ı q iii C .k.1 (cid:0) a k q (cid:0) q k (cid:0) / C k /. k ı q k C hhh g.q (cid:0) q /; ı q iii /; (cid:0) k q (cid:0) q k (cid:0) hhh g.q (cid:0) q /; ı q iii C k q (cid:0) q k (cid:0) hhh g.q (cid:0) q /; ı q iii . k ı q k C hhh g.q (cid:0) q /; ı q iii / C .k.1 (cid:0) a k q (cid:0) q k (cid:0) / C k /.2 hhh g.ı q/; ı q iii C hhh g.q (cid:0) q /; ı q iii //: .243/ If q D q and k q (cid:0) q k D akk C k ; .244/ then k.1 (cid:0) a k q (cid:0) q k (cid:0) / C k D and X .q; ı q; ı q; ı q/ D .0; 0; 0/; .246/ and the stability test is inconclusive. N
11. Further criteria.
Example 14.
We will apply the fourth order stability criterion to the discrete model of the buckling ofa rod of Example 10.The set L D n .q; ı q; ı q; ı q; ı q/ Q z V V V V I q L ; hhh g.u/; ı q iii D hhh g.u/; ı q iii D hhh g.u/; ı q iii D hhh g.u/; ı q iii D hhh g.u/; ı q iii D o .247/ is the fourth tangent set of L .The fourth order work function at q D q and k q (cid:0) q k D akk C k ; .248/ assumes the form X W L ! R W .q; ı q; ı q; ı q; ı q/ ‘ (cid:0)
0; 0; 0; 3.ka/ (cid:0) .k C k / k ı q k : .249/ The inequality X .q; ı q; ı q; ı q; ı q/ D (cid:0)
0; 0; 0; 3.ka/ (cid:0) .k C k / k ı q k > .0; 0; 0; 0/ .250/ for each .q; ı q; ı q; ı q; ı q/ L is the sufficient condition for stable equilibrium at q D q and k q (cid:0) q k D akk C k : .251/ N Reference.