On the non-integrability and dynamics of discrete models of threads
NNoname manuscript No. (will be inserted by the editor)
On the non-integrability and dynamicsof discrete models of threads
Valery Kozlov · Ivan Polekhin
Received: date / Accepted: date
Abstract
In the paper, we study the dynamics of pla-nar n -gons, which can be considered as discrete modelsof threads. The main result of the paper is that, un-der some weak assumptions, these systems are not inte-grable in the sense of Liouville. This holds for both com-pletely free threads and for threads with fixed pointsthat are placed in external force fields. We present suffi-cient conditions for the positivity of topological entropyin such systems. We briefly consider other dynamicalproperties of discrete threads and we also consider dis-crete models of inextensible yet compressible threads. Keywords
Inextensible thread · Non-integrability · Topological entropy · Discrete model · Planar linkage
The study of the motion of a flexible inextensible threadhas historically been a popular mechanical problem (see,for instance, the classical book by P. Appell [1]). Theimportance of this topic stems from the role that threads,tethers, ropes, chains and other string-like elements playin a wide range of industrial applications. During mostof the twentieth century, these elements were commonlyused in light and marine industry and in instrument en-gineering [2,3,4,5,6,7,8,9]. Later, in the second half ofthe twentieth century, it became understood that teth-ers can be also useful in space applications [10,11,12,13,
V. KozlovSteklov Mathematical Institute of Russian Academy of Sci-ences, Moscow, RussiaE-mail: [email protected]. PolekhinSteklov Mathematical Institute of Russian Academy of Sci-ences, Moscow, RussiaE-mail: [email protected] n ) of rigid segments of equal lengths l ;the segments form a broken line and are connected byplanar hinges; and, masses m , . . . , m n +1 are located inthe endpoints of the segments. It is possible to considerthreads with fixed points and threads that form closedcontours. In the latter case, the first mass point alwayscoincides with the last one. We will also briefly considerdiscrete models of compressible threads, which will bespecified below.Our model of an incompressible thread could becalled ‘a Poinsot thread’ after Louis Poinsot, who pro- a r X i v : . [ n li n . C D ] S e p Valery Kozlov, Ivan Polekhin posed to consider rigid balls strung on a thread to givean interpretation for negative tension that may occurin some equilibrium configurations of a flexible thread[1]. As mentioned earlier, it is possible to consider vari-ous constraints imposed on a thread. For instance, onecan fix some points of the system. The following fiveconfigurations can be considered as basic from the pointof view of possible applications:( i ) A thread with fixed endpoints (broken line withfixed points),( ii ) A closed thread with a fixed point ( n -gon with afixed point),( iii ) A closed thread (planar n -gon),( iv ) A thread with one fixed endpoint ( n -link pendu-lum),( v ) A free non-closed thread.All these systems can be considered as free, in thesense that there are no external forces acting on thesystem, and as threads in external potential force fields.One can also assume that there are internal forces act-ing between the masses or the segments of the thread.For instance, these forces can model various elastic prop-erties of our mechanical system.The topology of the configuration space of the cor-responding discrete system will play a key role in ourconsiderations. For cases ( iv ) and ( v ), the structure ofthe configuration space can be easily understood: it iseither an n -dimensional torus, or a direct product of an n -dimensional torus and a group of parallel translationsof the plane (which is isomorphic to R ). When thesesystems move by inertia (i.e., the only forces acting onthe system are the forces of reaction), we have naturalNoetherian first integrals. In case ( iv ), this first inte-gral is the kinetic moment w.r.t. the fixed point. Theconfiguration space obtained after the corresponding re-duction is an ( n − v ), thegroup of symmetries coincides with the symmetries ofthe Euclidean plane and we have three Noetherian inte-grals. After the reduction, we again obtain a system onan ( n − i ) — ( iii ),that will be our main objects of study, the topology ofthe configuration space can be more complex.The rest of this paper is structured as follows. First,we recall some results on the integrability of Hamilto-nian systems and we present auxiliary results concern-ing the topology of planar n -gons. In the next section,we show that threads described by models ( i ) — ( iii )are not integrable in the class of real analytical func-tions provided that some natural assumptions hold. Wealso present some geometrical results concerning thedynamics of these systems. Then, we discuss the ques-tion of the positivity of the topological entropy for our models. In the conclusion, we briefly consider severalrelated problems, including possible generalization ofour results to higher dimensions and possible modelsfor compressible threads and their properties. S, ω, H ), where S is a 2 n -dimensional smoothmanifold, ω is a symplectic structure on S and H : S → R is a smooth function, is called a Hamiltonian system.A smooth function F : S → R is called a first integralof the system ( S, ω, H ) if { F, H } ≡ . Here {· , ·} is the Poisson bracket corresponding to thesymplectic structure. We say that system ( S, ω, H ) isLiouville integrable (or simply, integrable) if1. There are n first integrals F = H, . . . , F n : S → R ;2. These functions are independent, that is, almost ev-erywhere on S , 1-forms dF , . . . dF n are linearly in-dependent;3. { F i , F j } ≡ i and j .Everywhere below we will consider analytic Hamilto-nian systems: manifold S is an analytic manifold, H is a real analytic function. The system is analyticallyintegrable if all of the functions F i are analytic.In our considerations, we will assume that S is acotangent bundle of an n -dimensional manifold M , thatis, S = T ∗ M . By q we will denote local coordinates on M and by p we denote local coordinates on T q M . Inparticular, in these coordinates the Poisson bracket hasthe standard form { F, G } = n (cid:88) i =1 (cid:18) ∂F∂q i ∂G∂p i − ∂F∂p i ∂G∂q i (cid:19) . A more detailed exposition of Hamiltonian mechanicscan be found, for instance, in [29,30]. The problem ofintegrability of Hamiltonian systems is discussed in de-tail in [31].The main tool that will be used here to prove thenon-integrability of our system is the following theoremof I.A. Taimanov on the non-integrability of geodesicflows [32,33].
Theorem 1
Given a geodesic flow on an n -dimensionalclosed analytic manifold M with an analytic Hamilto-nian function H : M → R . If dim H ( M, Q ) > n, (1) then there are no functions F , . . . , F n : M → R suchthat for some energy level F = H = const > we have n the non-integrability and dynamics of discrete models of threads 3
1. Functions F , . . . , F n are analytic and { F i , F j } = 0 for any i, j in a neighborhood of the level set H =const ;2. Differentials dF , . . . , dF n are linearly independenton H = const . As a corollary from this theorem, we obtain sufficientconditions for non-integrability of so-called natural Hamil-tonian systems. Let us recall that the system is callednatural if its Hamiltonian has the form H ( p, q ) = H ( p, q ) + H ( q ) = n (cid:88) i,j =1 g ij ( q ) p i p j + H ( q ) , (2)where H is a positive definite quadratic form in p (ki-netic energy). In accordance to the Maupertuis prin-ciple, projections of the solutions of this system onto M can be considered as geodesics of the Jacobi met-ric. To be more precise, consider level set H ( p, q ) = h ,where h > max H . The trajectories of the system withthe Hamiltonian function H on the level H = h thencoincide with the trajectories of the system with theHamiltonian function ˜ H ˜ H = n (cid:88) i,j =1 g ij ( q ) h − H ( q ) p i p j on the level set ˜ H = 1. Moreover, if the original Hamil-tonian system has a first integral F on a level set H = h ,then the corresponding geodesic flow of the Jacobi met-ric has a first integral ˜ F on T ∗ M (possibly, except forthe set ˜ H = 0) and˜ F ( p, q ) = F p (cid:113) ˜ H ( p, p ) , q . Corollary 1
Given an analytic manifold M such thatcondition (1) holds and a natural Hamiltonian systemon T ∗ M . Then, this system cannot be analytically in-tegrable. Indeed, if we have n first analytic independent integrals F , . . . , F n for the natural Hamiltonian system, then weobtain functions ˜ F , . . . , ˜ F n satisfying the conditions ofTheorem 1.Theorem 1 can also be applied to more general Hamil-tonian systems. Let us have a system with an analyticHamiltonian function HH = H ( p, q ) + H ( p, q ) + H ( q ) , (3)where H and H coincides with the corresponding termsin (2) and H ( p, q ) = n (cid:88) i =1 b i ( q ) p i . We will say that system (3) is integrable in the class ofpolynomial in p first integrals with independent highestdegree terms if there exist n first integrals F i : T ∗ M → R ( F = H ) of the form F i ( p, q ) = F m i i ( p, q ) + F m i − i ( p, q ) + . . . F i ( q ) , where F m i i , F m i − i , . . . F i are homogeneous in p ana-lytic polynomials of degrees m i , m i − , . . . { F m i i , F m j j } ≡ i and j , and dF m , . . . dF m n n are linearly independent almost everywhere. Corollary 2
Given an analytic manifold M such thatcondition (1) holds and a Hamiltonian system on T ∗ M with the Hamiltonian function (3) . Then, this systemcannot be integrable in the class of polynomials in p with independent highest degree terms. The proof directly follows from the fact that thehighest degree terms F m = H , . . . F m n n are first inte-grals for the geodesic flow with Hamiltonian ˜ H = H .It is worth mentioning here that all of the known in-tegrable mechanical systems are integrable in the classof polynomial first integrals with independent highestdegree terms.2.2 Topology of linkagesIn this section, we present the results on the topologicalproperties of planar linkages (see, for instance, [34,35]).Let us have n planar segments with lengths l , l ,..., l n , which form a closed polygon. In the followingwe assume that for the lengths the following conditionholds n (cid:88) i =1 l i ν i (cid:54) = 0 , for any ν i = ± . (4)We will denote the configuration space of the poly-gon, viewed up to isometries of the Euclidean plane, by˜ M :˜ M = { ( u , ..., u n ) ∈ S × ... × S : n (cid:88) i =1 l i u i = 0 } /SO (2) . (5)Equivalently, we can consider our n -gon with one ofits sides fixed: the configuration space of this systemnaturally coincides with ˜ M . Theorem 2 ˜ M is an analytic closed orientable mani-fold of dimension n − . Valery Kozlov, Ivan Polekhin
Remark 1
When condition (4) does not hold, there area finite number of singularities on ˜ M that correspondto the collinear configurations of the linkage. Note thatthis condition holds for a generic set of lengths. Definition 1
Given a polygon with lengths l , l , ..., l n ,we call a subset of its sides J = { i , i , ..., i k } short when (cid:88) i ∈ J l i < (cid:88) i (cid:54)∈ J l i . In the following, we will use the following result onthe topology of the configuration space of a linkage.
Theorem 3
Given a planar polygon satisfying (1) , let l i be a side of the maximal length (i.e., l i (cid:62) l j for any j ), for every k ∈ { , , ..., n − } , the homology group H k ( M l ; Z ) is a free Abelian group of rank a k + a n − − k ,where a k denotes the number of short subsets of k + 1 elements containing l i . Corollary 3
Let n = 2 r +1 and for all i we have l i = 1 .Then b k ( ˜ M ) = C kn − , for k < r − , C r − n − , for k = r − ,C k +2 n − , for k > r − . (6)The following result on the fundamental groups ofplanar polygons will be used when we will discuss topo-logical entropy. The proof can be found in [36] (see also[37], where the same technique has been used to calcu-late the fundamental groups for more complex types ofplanar linkages). Theorem 4
Let us have a planar polygon and either l i = 1 for all i , or l i = 1 for (cid:54) i (cid:54) n − and l n = l ,where (cid:54) l < n − , n (cid:62) and condition (4) holds.Then π ( ˜ M ) ∼ = (cid:42) a , . . . , a n − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a k , if { k, n } is not short [ a i , a j ] , if { i, j, n } is short (cid:43) (7) where [ a i , a j ] = a − i a − j a i a j . In other words, we have a free group with generators a , . . . , a n − and we put a k = 1 if { k, n } is not shortand we put [ a i , a j ] = 1 if { i, j, n } is short. In this section we study the non-integrability of mod-els ( i )–( iii ). For each of these models, we consider twoclasses of systems, which are different from the dynam-ical point of view: 1. A natural Hamiltonian system. In this case the Hamil-tonian of the corresponding thread has the form H = H ( p, q ) + H ( q ) , (8)where H ( p, q ) is a quadratic positive definite formin p , i.e., H is the kinetic energy; p = ( p , . . . p k )are the generalized momenta and q = ( q , . . . , q k )are the local coordinates on the configuration spaceof dimension k and H ( q ) is a function on the con-figuration space which corresponds to the potentialforces acting on the system. These forces includeboth external and internal forces, i.e., for instance,it can be an external force field of gravity or restor-ing forces caused by the springs located in the jointsof the thread. A free thread moving by inertia canbe considered as a natural system such that H ≡ H = H ( p, q ) + H ( p, q ) + H ( q ) , (9)where H ( p, q ) = (cid:80) ki =1 b i ( q ) p i and H and H aredefined as above. The term H corresponds to so-called gyroscopic forces. For instance, it can be mag-netic forces acting on the thread.3.1 A thread with fixed endpointsConsider the following natural model of a thread withfixed endpoints: a collection of n rigid planar segmentsof the same length, the segments are pairwise connectedby joints at their endpoints and form a planar brokenline between two fixed points. Without loss of the gener-ality we can assume that all segments have unit length.We will denote the distance between the fixed pointsby l and assume that l < n and l / ∈ N . Then (4) holdsfor the closed ( n + 1)-gon such that l = . . . l n = 1 and l n +1 = l (Fig. 1).The above specifies the kinematics of the thread. Itsdynamical behavior is determined by the distributionof mass of the thread and by the forces acting on thesystem.We assume that all mass of the thread is concen-trated in the joints, i.e., there are n − m i > i -th mass is located in the joint betweenthe i -th and the ( i + 1)-th segments of the broken line.We do not consider mass points at the fixed points sincethey do not affect the dynamics of the system. Whenthe thread is homogeneous it is natural to assume that m i = m j for all i and j .Everywhere below we assume that the thread moveswithout friction and the system is Hamiltonian. Let the n the non-integrability and dynamics of discrete models of threads 5 Fig. 1
An example configuration of an inextensible and in-compressible thread with fixed endpoints (model ( i )). system have k degrees of freedom, that is, the dimensionof the configuration space equals k .Consider the closed polygon such that one of itssides has length l and all other sides are of unit length.From Theorem 2 we obtain that k = n −
2. Therefore,we assume that n (cid:62)
4, since all one-dimensional casesare trivially integrable.Let us now calculate the first Betti number of thisspace.
Proposition 1 If < l < , then b ( M ) = (cid:40) n + 4 , if n = 4 , n, if n > . If < l < n − , l / ∈ N , then for n > we have b ( M ) = n and for n = 4 we have b ( M ) = 8 . If n − < l < n then b ( M ) = 0 .Proof The proof is a direct calculation based on The-orem 2. If 0 < l < n = 4, the total numberof elements in the thread is 5, we have a = 4 and b = a + a = 8. Similarly, if 0 < l < n = 4we have a = 5 and a = 4, b = a + a = 9. If1 < l < n − n = 4, then a = 4 and b = 2 a . If1 < l < n − n >
4, then a = n and a n − = 0, b = n . Finally, for n − < l < n we obtain b = 0.We see that (1) holds for all n and for all l < n − Proposition 2
Let < l < n − and l / ∈ N , and theHamiltonian function (8) of the system is an analyticfunction, then the system is not analytically integrable.Proof Indeed, dim( M ) = n − b ( M ) (cid:62) n .Similarly, from Corollary 2, we have Proposition 3
Let < l < n − and l / ∈ N , and theHamiltonian function (9) of the system is an analyticfunction, then the system is not integrable in the class ofpolynomials in p with independent highest degree terms. In other words, the system of a discrete thread be-tween two fixed points cannot be analytically integrablefor l < n −
2. This holds for a free thread and for athread in external or internal force fields. In particular,if we have a thread in a gravity field, then this system isnot analytically integrable. If we add a magnetic forcesto the system, then this system cannot be integrated inthe class of polynomials in p with independent highestdegree terms.Note that for large n , that is, when the discretemodel of a thread is relatively fine, the condition l 1, then analytic Hamiltoniansystem with configuration space M × K cannot be in-tegrable. Indeed, from the K¨unneth theorem, we have b ( M × K ) = b ( M ) + b ( K ) (cid:62) n + k − . Obviously, dim( M × K ) = n + k − k -link pendulum attached to one ofthe moving joints of the thread. The configuration spaceof the pendulum is a k -dimensional torus and b ( T k ) = k . In conclusion of the section, we present a geometri-cal result concerning the dynamics of the thread in the Valery Kozlov, Ivan Polekhin most general case, that is, in the presence of potentialand gyroscopic forces.First, we shortly recall the correspondence betweenthe Hamiltonian and Lagrangian approaches to the dy-namics of mechanical systems. Given a Hamiltonianfunction of the form (9), we can obtain a Lagrangian L by means of the Legandre transformation: L ( q, ˙ q ) = ˙ q · p − H ( p, q ) , ˙ q = ∂H∂p . In the new variables ( q, ˙ q ) we have L ( q, ˙ q ) = L ( q, ˙ q ) + L ( q, ˙ q ) + L ( q ) , (10)where, again, L ( q, ˙ q ) is a quadratic positive definiteform in ˙ q and L ( q, ˙ q ) is linear in ˙ q . The dynamics onthe tangent bundle T M is defined by the correspondingLagrange equations. Proposition 4 Given a Lagrangian system with La-grangian (10) and an energy level h > max M ( − L ) , thenany two configurations q , q ∈ M of the thread can beconnected by a solution with energy h provided that h + L ) L > L (11) for all ( q, ˙ q ) on the corresponding energy level.Proof In accordance to the Maupertuis principle, a path γ : [ t , t ] → M is a trajectory of a solution of the La-grangian system iff γ ( t ) is a critical point for the func-tional FF ( γ ) = t (cid:90) t (2 (cid:112) ( h + L ( γ )) L ( γ, ˙ γ ) + L ( γ, ˙ γ )) dt in the class of all paths of fixed energy h [29]. If in-equality (11) holds, then F defines a Finsler length on M [39,40]. From the Hopf-Rinow theorem for Finslermanifolds [39,40], we have that any two points of M can be connected by a Finsler geodesic. This geodesiccorresponds to the desirable solution.3.2 Closed threadsIn this section we will consider models ( ii ) and ( iii ). Toa large degree they are similar and both these modelswill be shown to be non-integrable. However, for model( iii ), we will impose some additional conditions to provethe non-integrability.Consider a closed n -gon assuming that all its sideshave the same unit length and n is an odd number n (cid:62) 5. Then (4) obviously holds. We also assume thatone of the points of the n -gon is fixed and there are n − m i located in all non-fixed vertices of the n -gon (Fig. 2). Note that we allow self-intersections dur-ing the motion of the n -gon. Similarly to the case of athread with two fixed points, we assume that all forcesacting on the system are potential and the system isHamiltonian with the Hamiltonian of the form (8). Proposition 5 Let the Hamiltonian function (8) of thesystem be an analytic function. Then the system is notanalytically integrable.Proof The configuration space M of the system is thedirect product of a one-dimensional circle and ˜ M andhas dimension n − 2. From Corollary 3 we obtain that b ( ˜ M ) = n − 1. Therefore, b ( M ) = b ( ˜ M ) + 1 = n andwe can apply Theorem 1.Similar result holds for the system with gyroscopicforces. Proposition 6 Let the Hamiltonian function (9) of thesystem be an analytic function. Then the system is notintegrable in the class of polynomials in p with indepen-dent highest degree terms. Again, the system has the Hamiltonian function ofthe form (8) when we consider a totally free thread,that is, there are no external or internal forces actingon the system, except for the forces of reaction. Also,we can consider a thread in an external force field or athread with interactions between its elements.Let us now consider a closed thread without a fixedpoint, that is, a planar n -gon with the sides of unitlength and n masses m i located in the vertices. Fig. 2 An example configuration of an inextensible and in-compressible thread with a fixed point (model ( ii )). The configuration space of this system is not com-pact and Theorem 1 cannot be applied directly. How-ever, if we assume that there are no external forces act-ing on the thread, we can consider the reduced system n the non-integrability and dynamics of discrete models of threads 7 with a compact configuration space. To be more precise,let x and y be the Cartesian coordinates of some masspoint of the thread and we consider these coordinatesas a part of the set of generalized coordinates. Let theHamiltonian function of the system has the form (8).Since there are no external forces acting on the system,we can conclude that H does not depend on x and y .Clearly, ∂H∂x = c x = const , ∂H∂y = c y = const . After the Routh reduction w.r.t. variables x and y weobtain a Hamiltonian system with the Hamiltonian ofthe form (9) where H ≡ c x = 0 and c y = 0.Therefore, similarly to Propositions 2 and 5, we obtain Proposition 7 Let us consider a closed n -gon movingon a plane without friction. The lengths of the sides ofthis n -gon equal ( n is an odd number greater than )and masses m i are located in the vertices of the polygon.Suppose that the only forces acting on the system are theforces of reaction and internal potential forces actingbetween the elements of the thread. Let c x = 0 , c y = 0 and the Hamiltonian function of the reduced system isan analytic function of the form (8) . Then the reducedsystem is not analytically integrable. If the initial system contains non-zero terms H orat least one of the constants c x or c y does not equalzero, then the Hamiltonian of the reduced system takesthe form (9). Therefore, we obtain the following result. Proposition 8 Let us consider a closed n -gon movingon a plane without friction. The lengths of the sides ofthis n -gon equal ( n is an odd number greater than )and masses m i are located in the vertices of the polygon.Suppose that the Hamiltonian function H of the systemis an analytic function and has the form (9) and H does not depend on x and y . Then the reduced systemis not integrable in the class of polynomials in p withindependent highest degree terms. i ). Let l i > 0, 1 (cid:54) i (cid:54) n be the lengths of segments and l > b ( M ) (cid:62) n − b ( M ) (cid:62) a . Therefore, if a (cid:62) n − 1, then thesystem is not integrable. Let l or l j (for some 1 (cid:54) j (cid:54) n )be the side of the maximal length. If there are at least n − l i k , 1 (cid:54) k (cid:54) n − l i k and l j (or l ) is a short subset, then the corresponding systemis not integrable in the sense of Propositions 2 and 3.Absolutely similar conditions can be formulated formodels ( ii ) and ( iii ). For these cases we have to obtain b ( ˜ M ) (cid:62) n − n is the number of segments inthe thread. Therefore, there should be at least n − l i k , 1 (cid:54) k (cid:54) n − l j ) such that the pair of lengths l i k and l j is ashort subset. First, let us recall the definition of the topological en-tropy (see, for instance, [41]). Let X be a compact met-ric space with a metric d and f : X → X be a continu-ous map. Consider the following sequence of metrics d n ( x, y ) = max (cid:54) i (cid:54) n − d ( f i ( x ) , f i ( y )) . Consider an open ball B ( x, ε, n ) = { y ∈ X : d n ( x, y ) <ε } . A set U ⊂ X is an ( n, ε )-covering if X ⊂ (cid:83) x ∈ E B ( x, ε, n ).Let S ( ε, n ) be the minimal number of elements in an( n, ε )-covering. Put h ( f, ε ) = lim sup n →∞ n log S ( f, ε, n ) . Then, the topological entropy of the map f is definedas h ( f ) = lim ε → h ( f, ε ) . The definition of the topological entropy for flows canbe expressed in terms of the topological entropy formaps: let us have a flow ϕ t : R × X → X , then we put f = ϕ . Remark 2 This definition is based on a metric structureon X . However, it can be shown that this definition doesnot depend on the choice of the metric, provided thatall metrics define the same topology on X . A definitionthat is not based on the metric structure has been givenin [42]. The definition given above was first given in [43].In addition, the first definition of entropy for a dynam-ical system has been formulated by A.N. Kolmogorov[44]. Valery Kozlov, Ivan Polekhin For a geodesic flow on a Riemannian manifold thetopological entropy can be defined as follows [45]: h = lim L →∞ L log (cid:90) M × M n L ( x, y ) dxdy, where n L ( x, y ) is the number of geodesics of lengths nomore than L connecting points x and y of manifold M .Positivity of the topological entropy usually corre-sponds to the complexity of the dynamics of a system.It can also imply the chaotic behavior of a system [46].At the same time, the positivity of topological entropyis not equivalent to the ergodicity and there are non-ergodic systems with a positive topological entropy.Let us have a geodesic flow on a closed Rieman-nian manifold. It is known that for some manifolds itis impossible to find a metric with zero topological en-tropy, that is, for any given smooth metric, the entropyis positive. For instance, if the fundamental group ofthe manifold is a group of exponential growth, then thetopological entropy of the geodesic flow is positive. Thedetails can be found in [43,47], where the problem ofexistence of a metric with zero entropy is considered.In addition, the following has been proven in [43]. Theorem 5 If there exists a metric of negative sec-tional curvature on a closed manifold, then the geodesicflow on this manifold has a positive topological entropyfor any metric. It is known that there exists a metric of negativecurvature on any two-dimensional closed manifold ofgenus greater than one [48].In particular, for the previous discrete models ofthreads, the topological entropy can be proven to bepositive when the thread is moving by inertia. To bemore precise, given a thread with two fixed endpointsand n = 4, the dynamics is described by the geodesicequation provided the motion of the thread is free (i.e.,the only forces acting on the thread are the forces ofreaction). The metric is given by the kinetic energy ofthe system and the genus of the configuration manifoldis greater than one. Therefore, the topological entropyis strictly positive.Similar result holds for model ( iii ). However, it isworth mentioning that results about the positivity ofgeodesic flows can only be applied here for the caseswhere the constants of the Noetherian integrals equalzero.To be more precise, we can conclude that the fol-lowing results hold for two-dimensional configurationspaces. Proposition 9 Consider a thread with fixed endpoints.Let n = 4 and l i = 1 for all i . Let the distance between the fixed points be l < and condition (4) holds. Sup-pose that there are massive points with masses m i lo-cated in the joints of the thread and that the only forcesacting on the system are the forces of reaction. Then,the topological entropy of this system is positive.Proof It is known that the genus g of the surface equals b / 2, that is, for our surface we have g = 4. Proposition 10 Consider a closed thread. Let n = 5 and l i = 1 for all i . Suppose that there are five mas-sive points with masses m i located in the joints of thethread and that the only forces acting on the system arethe forces of reaction. Also suppose that the constantsof three Noetherian first integrals equal zero. Then, thetopological entropy of the system (after the Routh re-duction) is positive. Some results on the existence of a metric corre-sponding to zero topological entropy for low dimen-sional manifolds can be found in [49,50,51].In particular, it was proven in [51] that, given a four-dimensional closed manifold M with an infinite funda-mental group, it is only possible to find a metric on M with zero topological entropy when the Euler charac-teristic of M is zero.As a corollaries from this result, we obtain the fol-lowing. Proposition 11 Consider a thread with fixed endpoints.Let n = 6 and l i = 1 for all i . Let the distance betweenthe fixed points is l < , l / ∈ N . Suppose that there aremassive points with masses m i located in the joints ofthe thread and the only forces acting on the system arethe forces of reaction. Then, the topological entropy ofthis system is positive.Proof Consider the case when 1 < l < 4. For the Eulercharacteristic we have χ = b − b + b − b + b and b = b , b = b . Therefore, b = a + a = 1, b = a + a = 6 and b = 2 a = 30 (for 1 < l < 2) or b = 0(for 2 < l < χ (cid:54) = 0. The case 0 < l < iii ). Proposition 12 Consider a closed thread. Let n = 7 and l i = 1 for all i . Suppose that there are massivepoints with masses m i located in the joints of the threadand the only forces acting on the system are the forcesof reaction. Also suppose that the constants of threeNoetherian first integrals equal zero. Then, the topolog-ical entropy of the system (after the Routh reduction)is positive.Proof From Corollary 3, we have χ = 2(1 − C ) (cid:54) = 0. n the non-integrability and dynamics of discrete models of threads 9 Note that the fundamental groups of these systemsare clearly infinite because their abelianizations, thefirst homology groups, are infinite.Note that a result similar to Proposition 12 holdsfor model ( ii ) if we assume that the only Noetherianfirst integral equals zero. However, if we do not wantto consider the reduced system and, at the same time,we want to obtain a configuration space of dimension 4,then there should be six segments in the closed contour.If we assume that these segments are of the same length,then the configuration space will not be a smooth man-ifold.For an arbitrarily large n (i.e., for the cases whenthe thread is modeled by a large number of segments),it is also possible to prove that the entropy is positivebased on Theorem 4. Proposition 13 Consider a thread with fixed endpoints.Let n > and l i = 1 for all i . Let the distance betweenthe fixed points is l and n − < l < n − , l / ∈ N . Supposethat there are massive points with masses m i located inthe joints of the thread and that the only forces act-ing on the system are the forces of reaction. Then, thetopological entropy of this system is positive.Proof From Theorem 4 we have that π ( M ) is free with n − { i, n + 1 } is always short and, con-versely, { i, j, n + 1 } is never short. Hence, π ( M ) is afree group, that is, a group of an exponential growth.Note, that for the closed thread with equal segmentsfrom Theorem 4, we obtain that π ( ˜ M ) is commuta-tive. Therefore, we cannot conclude that the entropy ispositive. Nevertheless, one can expect the topologicalentropy to be positive for these systems as well, yet theproof of this fact should follow not from the topologi-cal properties of the configuration space, but from themetric properties defined by the distribution of mass ofthe thread. To the best of our knowledge, the above propositionsgive the first non-trivial applications of the Taimanov’stheorem [32,33]. Note that, apparently, models ( iv ) and( v ) are also non-integrable. However, again, this non-integrability does not follow from the topological prop-erties of the configuration space (( n − n -gon where n = 2 k + 1and all l i = 1, odd Betti numbers of the configurationspace (again, considered up to the symmetries of theEuclidean space) vanish. Therefore, Theorem 1 cannotbe applied and, similarly to models ( iv ) and ( v ), non-integrability does not follow from these topological con-siderations. Here it is worth mentioning that there is aconjecture [53] generalizing Theorem 1 that claims thatthe system is not integrable if for some k dim H k ( M, Q ) > C kn . If this conjecture is true, that it is also possible to provethe non-integrability of spatial threads.It is also possible to consider another type of thread,a thread that is inextensible yet can be compressed. Inthis case one should assume that the distance betweentwo consecutive mass points is not equal to l i , but doesnot exceed this value. From the mechanical point ofview, one can imagine that the mass points are con-nected not by rigid massless rods, but by inextensibleropes.In the simplest case when we have only one masspoint connected to two fixed points. The motion is as-sume to be free, that is, there are no external forcesacting on the system. Let L be the distance between thefixed points, l and l be the lengths of the two ropesconnecting the mass point to the fixed points (Fig. 3).This system can be considered as a billiard witha non-smooth boundary. One of the first works wherethis system was considered for l = l is [54]. Later thiscase was studied numerically [55,56]. It was shown thatfor almost all distances between the fixed points, thesystem is not ergodic, since there exist stable periodictrajectories. Apparently, for l (cid:54) = l , the ergodicity ofthe corresponding billiard systems is not exceptional.To be more precise, in the two-dimensional space ofparameters l /l , L there is a set of non-zero measurecorresponding to the ergodic systems [57]. This set is asubset of all systems with the hyperbolic periodic tra-jectory of period 2 (this trajectory corresponds to thehorizontal periodic motion in Fig. 3). Note, that thestability of the elliptic trajectory of period 2 has beenrigorously established in [58], of course, these systemscannot be ergodic.The dynamics of two and more mass points con-nected by inextensible ropes is even more complex and,to the best of our knowledge, has not been studied —at least numerically — before.However, it is possible to obtain a geometric resultconcerning the dynamics of the compressible threadsprovided that we consider ‘almost inextensible’ threads.Let us have n massive points moving on a plane withoutfriction and the first and the last points are fixed. We Fig. 3 An example configuration of a simple model of aninextensible and compressible thread with two fixed points. assume that the point with number 2 (cid:54) i (cid:54) n − i − i + 1 and the potential energyof this interaction has the form U ( r i − ,i ) + U ( r i,i +1 ),where r i − ,i and r i,i +1 are the distances between thecorresponding points and U ( d ) is a smooth monotonousfunction such that U ( d ) ≡ d (cid:54) U ( d ) → + ∞ as d → + ∞ . Let the Lagrangian of the system havethe form (10), that is, we assume that there can beexternal potential and gyroscopic forces acting on thesystem. Since the total energy L − L = h , where h ∈ R , does not change along the solutions of the con-sidered system and L (cid:62) 0, then for any solution wehave − L ( q ) + h (cid:62) 0. Therefore, for a given energy h ,the possible motion area B h is defined as follows B h = { q : − L ( q ) + h (cid:62) } . (12)If U ( d ) is a rapidly increasing function, then, for a fixed h , the maximum distance between any two consecutivepoints is close to 1, that is, the thread is ‘almost inex-tensible’. The following result is proved in [59] Theorem 6 Let B h be a compact region and there areno critical points of L at boundary ∂B h . If the in-equality h + L ) L > L is true in B h \ ∂B h for any ˙ q (cid:54) = 0 , then any point inside B h can be connected withthe boundary ∂B h by a solution of energy h . This result in some sense complements Proposition4: we obtain that any configuration of the thread inthe possible motion area can be obtained if we startfrom the boundary ∂B h . In particular, if L ≡ 0, thepotential energy of the external forces acting on thesystem is bounded and h is relatively large, then we canconclude that any configuration in B h can be obtainedfrom another configuration such that at least one pairof massive points are under tension (the correspondingdistance is slightly greater than 1).In conclusion, returning to the question of non-integrability,we would like to mention an interesting parallel between the non-integrability of threads with inner interactionsbetween the elements, which can be considered as var-ious models for elastic properties of the system, andthe classical wave equation describing the motion of anelastic string with fixed endpoints. In contrast to ourmodel of the thread, this equation can be integrated ex-plicitly and the general solution is a sum of the standingwaves. The key difference between these two systems isthat the wave equation describes the motion of an ex-tensible string. Therefore, it may be useful to consideryet another model based on a planar or spatial polygonwith extensible sides. The topology of such systems hasbeen already studied in [60]. Acknowledgment This work was performed at the Steklov InternationalMathematical Center and supported by the Ministry ofScience and Higher Education of the Russian Federa-tion (agreement no. 075-15-2019-1614).