The Hojman Construction and Hamiltonization of Nonholonomic Systems

In this paper, using the Hojman construction, we give examples of various Poisson brackets which differ from those which are usually analyzed in Hamiltonian mechanics. They possess a nonmaximal rank, and in the general case an invariant measure and Casimir functions can be globally absent for them.

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Дата:	2016
Автори:	Bizyaev, I.A., Borisov, A.V., Mamaev, I.S.
Формат:	Стаття
Мова:	English
Опубліковано:	Інститут математики НАН України 2016
Назва видання:	Symmetry, Integrability and Geometry: Methods and Applications
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Цитувати:	The Hojman Construction and Hamiltonization of Nonholonomic Systems / I.A. Bizyaev, A.V. Borisov, I.S. Mamaev // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 41 назв. — англ.

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spelling	irk-123456789-1474352019-02-15T01:23:32Z The Hojman Construction and Hamiltonization of Nonholonomic Systems Bizyaev, I.A. Borisov, A.V. Mamaev, I.S. In this paper, using the Hojman construction, we give examples of various Poisson brackets which differ from those which are usually analyzed in Hamiltonian mechanics. They possess a nonmaximal rank, and in the general case an invariant measure and Casimir functions can be globally absent for them. 2016 Article The Hojman Construction and Hamiltonization of Nonholonomic Systems / I.A. Bizyaev, A.V. Borisov, I.S. Mamaev // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 41 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 37J60; 37J05 DOI:10.3842/SIGMA.2016.012 http://dspace.nbuv.gov.ua/handle/123456789/147435 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	In this paper, using the Hojman construction, we give examples of various Poisson brackets which differ from those which are usually analyzed in Hamiltonian mechanics. They possess a nonmaximal rank, and in the general case an invariant measure and Casimir functions can be globally absent for them.
format	Article
author	Bizyaev, I.A. Borisov, A.V. Mamaev, I.S.
spellingShingle	Bizyaev, I.A. Borisov, A.V. Mamaev, I.S. The Hojman Construction and Hamiltonization of Nonholonomic Systems Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Bizyaev, I.A. Borisov, A.V. Mamaev, I.S.
author_sort	Bizyaev, I.A.
title	The Hojman Construction and Hamiltonization of Nonholonomic Systems
title_short	The Hojman Construction and Hamiltonization of Nonholonomic Systems
title_full	The Hojman Construction and Hamiltonization of Nonholonomic Systems
title_fullStr	The Hojman Construction and Hamiltonization of Nonholonomic Systems
title_full_unstemmed	The Hojman Construction and Hamiltonization of Nonholonomic Systems
title_sort	hojman construction and hamiltonization of nonholonomic systems
publisher	Інститут математики НАН України
publishDate	2016
url	http://dspace.nbuv.gov.ua/handle/123456789/147435
citation_txt	The Hojman Construction and Hamiltonization of Nonholonomic Systems / I.A. Bizyaev, A.V. Borisov, I.S. Mamaev // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 41 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 12 (2016), 012, 19 pages The Hojman Construction and Hamiltonization of Nonholonomic Systems? Ivan A. BIZYAEV †‡, Alexey V. BORISOV †§ and Ivan S. MAMAEV † † Udmurt State University, 1 Universitetskaya Str., Izhevsk, 426034 Russia E-mail: bizaev 90@mail.ru, borisov@rcd.ru, mamaev@rcd.ru ‡ St. Petersburg State University, 1 Ulyanovskaya Str., St. Petersburg, 198504 Russia § National Research Nuclear University MEPhI, 31 Kashirskoe highway, Moscow, 115409 Russia Received October 05, 2015, in final form January 26, 2016; Published online January 30, 2016 http://dx.doi.org/10.3842/SIGMA.2016.012 Abstract. In this paper, using the Hojman construction, we give examples of various Pois- son brackets which differ from those which are usually analyzed in Hamiltonian mechanics. They possess a nonmaximal rank, and in the general case an invariant measure and Casimir functions can be globally absent for them. Key words: Hamiltonization; Poisson bracket; Casimir functions; invariant measure; non- holonomic hinge; Suslov problem; Chaplygin sleigh 2010 Mathematics Subject Classification: 37J60; 37J05 1 Introduction This paper is dedicated to Professor S. Benenti on the occasion of his 60th birthday. He has made an essential contribution to the development of various branches of mechanics and mathematics. In nonholonomic mechanics, which is the subject of this paper, he proposed a new construc- tive form of the equations of motion [1] and gave a new example of nonlinear nonholonomic constraint [2]. In this paper we consider the possibility of representing some well-known and new nonholo- nomic systems in Hamiltonian, more precisely, Poisson form. We note from the outset that such a representation is usually possible for reduced (incomplete) equations [14] and after rescaling time (conformal Hamiltonianity). Hamiltonian systems are the best-studied class of systems from the viewpoint of the theory of integrability, stability, topological analysis, perturbation theory (KAM theory) etc. There- fore, the investigation of the possibility of representing the equations of motion in conformally Hamiltonian form is an important, but poorly studied problem, which is usually called the Hamiltonization problem. There are various well-known obstructions to Hamiltonization, which are examined in detail in [7] (see also [5]). This problem has several aspects: local, semilocal, and global. In this sense the Hamiltonization problem is in many respects analogous to the problem of existence of analytic integrals and a smooth invariant measure (see [32]). In nonholonomic mechanics, the reducing multiplier method of Chaplygin [21] (developed further in [8]) provides a powerful tool for the Hamiltonization of equations of motion. However, this method does not apply to the problems presented below due to the fact that it is impossible to represent the equations of motion in the form of the Chaplygin system, or the examples considered exhibit everywhere the absence of a smooth invariant measure with density depending ?This paper is a contribution to the Special Issue on Analytical Mechanics and Differential Geometry in honour of Sergio Benenti. The full collection is available at http://www.emis.de/journals/SIGMA/Benenti.html mailto:bizaev_90@mail.ru mailto:borisov@rcd.ru mailto:mamaev@rcd.ru http://dx.doi.org/10.3842/SIGMA.2016.012 http://www.emis.de/journals/SIGMA/Benenti.html 2 I.A. Bizyaev, A.V. Borisov and I.S. Mamaev only on positional variables (as required by the Chaplygin method). Other Hamiltonization methods are examined in [18]. We note that the above-mentioned paper is of formal character and gives no nontrivial mechanical examples encountered in applications. Poisson brackets whose appearance cannot be explained by employing the Chaplygin method were obtained in [4, 6, 10, 38, 39] with the help of an explicit algebraic ansatz. In this paper it is shown that these brackets can be obtained using an algorithm that we call the Hojman construction (its foundations were laid down already by S. Lie). It is based on the existence of conformal symmetry fields, i.e., symmetry fields obtained after rescaling time, and was developed further in [20, 28]. The rank of the Poisson bracket obtained by using the Hojman construction is, as a rule, smaller than the dimension of the phase space; for it a smooth (or even singular) invariant measure and global Casimir functions may be absent in the general case, and the system’s behavior itself may differ greatly from Hamiltonian behavior, moreover, there may exist limit cycles. Therefore, the resulting Poisson brackets are of rather limited utility. In practice, such a Hamiltonian representation may turn out to be useless (especially for infinite-dimensional systems [20]). Indeed, the basic methods of Hamiltonian mechanics have been developed for canonical sys- tems. However, as shown in [3], the representation found by us can be useful for the investigation of stability problems. Problems of Hamiltonization and various aspects of the dynamics of nonholonomic systems are investigated in [6, 9, 11, 15, 16, 22, 23, 38, 39]. 2 Tensor invariants and Hamiltonization of dynamical systems 2.1 Basic notions and definitions Let a dynamical system be given on some manifold Mn, a phase space. ẋ = v(x), (1) where x = (x1, . . . , xn) are local coordinates, v = (v1(x), . . . , vn(x)). The evolution of an arbitrary (smooth) function onMn along the vector field (1) is given by the linear differential operator Ḟ = v(F ) = n∑ i=1 vi ∂F ∂xi . As is well known, the behavior of a dynamical system is in many respects determined by tensor invariants (conservation laws), from which one can, as a rule, draw the main data on its dynamics. For exact formulations see [12]. We only note that three kinds of tensor invariants are encountered particularly frequently: – the first integral F (x) for which Ḟ = v(F ) = 0; – symmetry field u(x) for which [v,u] = 0; The Hojman Construction and Hamiltonization of Nonholonomic Systems 3 – the invariant measure µ = ρ(x)dx1 ∧ · · · ∧ dxn whose density ρ(x) is everywhere larger than zero and satisfies the Liouville equation div(ρv) = 0, (2) where ∧ denotes the exterior product. In this paper, unless otherwise stated, all geometric objects (first integrals, vector fields etc.) are assumed to be analytical on Mn. Another important tensor invariant is the Poisson structure, which allows the equations of motion to be represented in Hamiltonian form. In applications, such a representation can usually be obtained only after rescaling time as dτ = N (x)dt (if the system is a priori not Hamiltonian), i.e., the equations of motion (1) are represented in conformally Hamiltonian form ẋ = N (x)J(x) ∂H ∂x , (3) where H(x) is a Hamiltonian that is a first integral, and J(x) = ‖Jij(x)‖ is a Poisson structure, i.e., a skew-symmetric tensor field satisfying the Jacobi identity n∑ l=1 ( Jlk ∂Jij ∂xl + Jli ∂Jjk ∂xl + Jlj ∂Jki ∂xl ) = 0, i, j, k = 1, . . . n, (4) and N (x) is a scalar function – a reducing multiplier. The Poisson structure J(x) allows one to define in a natural way the Poisson bracket of the functions f and g by the formula {f(x), g(x)} = n∑ i,j=1 Jij ∂f ∂xi ∂g ∂xj . Remark 1. Time rescaling cannot be applied at points where the reducing multiplier N (x) vanishes, hence, Hamiltonization is possible only on the set M̃n = {x ∈ Mn \| N (x) 6= 0}. As a result, the behavior of the trajectories in general (on the entire phase space Mn) can considerably differ from the behavior of the trajectory of Hamiltonian systems. In particular, not only tori, but also two-dimensional integral manifolds can be arbitrary in this case, see, for example, the Suslov problem [10]. In many examples the Poisson structure turns out to be rank J < n. As is well known, the entire phase space (in the domain of the constant rank J) is foliated by symplectic leaves O of dimension dimO = rank J. In the simplest case the symplectic leaves are given as the level surfaces of a set of global Casimir functions Oc = { x ∈Mn \|C1(x) = c1, . . . , Cm(x) = cm } , n∑ j=1 Jij(x) ∂Ck ∂xj = 0, i = 1, . . . , n, k = 1, . . . ,m = n− rank J. (5) Nevertheless, in most of the examples considered below the number of global Casimir functions turns out to be less than m. This may lead to unusual (from the viewpoint of the standard theory of Hamiltonian systems) behavior of trajectories of the system (3). We note that there are three levels of analysis of the problem of the existence of tensor invariants: 4 I.A. Bizyaev, A.V. Borisov and I.S. Mamaev – local level – in a neighborhood of a nonsingular point of the vector field (1), – semilocal level – in a neighborhood of invariant sets of the system (1), such as fixed points, periodic orbits, invariant tori etc., – global level – on the entire phase space Mn. From the local point of view, obstructions to the existence of any tensor invariants are not encountered by virtue of the rectification theorem for vector fields. Therefore, in what follows it is implied that we consider the system from the semilocal and global points of view. 2.2 The general Hojman construction Consider a conformally Hamiltonian system in the form (3). From the point of view of integra- bility, the case where rank J = 2 is regarded as the simplest case. As a rule, in this case it is assumed that the system (3) admits a natural (global) restriction to the two-dimensional symplectic leaf O2 and reduces to a Hamiltonian system with one degree of freedom, and its trajectories are the level lines of the restriction of the Hamiltonian H\|O2 . It turns out that such a conclusion cannot be drawn in the general case, when the properties of solutions (5) are unknown to us. As already noted, a full set of global Casimir functions can be absent in J (some of them are defined only locally). As a result, the symplectic leaf can be immersed in the phase space in a fairly complicated (in particular, chaotic) way, and the above-described picture is not always realized. Nevertheless, the above Poisson structure turns out to be useful for the investigation of stability problems [29]. Consider a Poisson structure J(x) of rank 2 on Mn for which there is a pair of (globally defined) vector fields v(x) and u(x) such that J(x) = v(x) ∧ u(x). (6) Remark 2. Every skew-symmetric bivector field of constant rank 2 locally admits such a rep- resentation. According to the Darboux theorem, the Jacobi identity (4) holds only in the case where the distribution given by these vector fields is integrable [v,u] = µ1(x)v + µ2(x)u. (7) The Casimir functions are simultaneously the integrals of the fields v and u: u(Ck) = v(Ck) = 0, k = 1, . . . ,m′. Remark 3. Condition (7) was known already to S. Lie; it can also be obtained if the Jacobi identity is rewritten by means of the Schouten bracket [[J,J]] = 0. Using the properties of the Schouten bracket, for the Poisson structure (6) we find [29]: [[J,J]] = 2[v,u] ∧ v ∧ u = 0, whence we obtain condition (7). It turns out that in many examples such decomposable Poisson structures allow one to Hamil- tonize in a natural way the dynamical systems in question. This approach was proposed in [20, 28]. We present the necessary results and omit the proofs, which are completely straight- forward. The Hojman Construction and Hamiltonization of Nonholonomic Systems 5 Theorem 1 (Hojman [28]). Suppose that the system (1) possesses a first integral H and a vector field u(x) such that [v,u] = µ(x)v, Hu(x) = u(H) 6≡ 0, (8) then the initial system (1) can be represented in conformally Hamiltonian form ẋ = H(−1) u (x)J(2)(x) ∂H ∂x , J(2)(x) = v ∧ u. (9) Remark 4. The above conformally Hamiltonian representation (9) is not defined for points x at which Hu(x) = 0, and the symbol 6≡ means that Hu(x) must not be equal to zero for any x. It follows from conditions (8) that the vector field u(x) is not tangent to the level surface H(x) = const and, moreover, v(Hu) = 0, hence, in this case the reducing multiplier Hu(x) is the first integral of the system. If, in addition, the system possesses additional tensor invariants, then a natural generalization of this result holds (see [28] for details). Proposition 1. Suppose that the system (1) satisfies the conditions of Theorem 1 and, moreover, possesses the symmetry fields u1 and u2 which define the integrable distribution and preserve the Hamiltonian [u1,v] = [u2,v] = 0, u1(H) = u2(H) = 0, [u1,u2] = λ1(x)u1 + λ2(x)u2, then the system (1) admits the conformally Hamiltonian representation ẋ = H(−1) u (x)J(4)(x) ∂H ∂x , J(4)(x) = v ∧ u+ u1 ∧ u2, rank J(4) = 4. Below we consider some nonholonomic mechanics problems illustrating the applicability of the above theorem and allowing one to obtain Poisson brackets with various (unusual) properties. The following construction shows that any dynamical system admits a rank 2 Hamiltonization in an extended phase space. Let v denote the vector field that defines the dynamics of a nonholonomic system on some manifold M. Let E : M → R be the energy of the system, so that v(E) = 0. Consider the following rank two Poisson structure in M× R: J = v ∧ ∂ ∂y , where y is the coordinate on the factor R of the Cartesian product M× R. The fact that J satisfies the Jacobi identity is obvious from the fact that [v, ∂∂y ] = 0. It is obvious that the Hamiltonian vector field of y with respect to is v. Moreover, E is a Casimir of J. This example shows how the symplectic leaves of a rank two Poisson structure can be im- mersed in the phase space in a rather complicated way. Moreover, it also illustrates that usually the construction of brackets in an extended phase space cannot add anything new to understand the properties of a given system. 6 I.A. Bizyaev, A.V. Borisov and I.S. Mamaev 2.3 Invariant measure in Hamiltonian systems Let us briefly discuss the problem of existence of an invariant measure in Hamiltonian systems on Mn with a Poisson structure ẋ = J(x) ∂H ∂x , rank J < n. (10) We recall that for a symplectic manifold the rank of J equals n = 2k and that, according to the Liouville theorem, an invariant measure exists for an arbitrary Hamiltonian. By analogy with this, if for the system (10) with an arbitrary Hamiltonian there exists the same invariant measure, it is called the Liouville measure. Its density satisfies the system of equations on Mn n∑ j=1 ∂ ∂xj (ρJij(x)) = 0, i = 1, . . . , n. A manifold with this Poisson structure J(x) and volume form ρ(x)dx1 ∧ · · · ∧ dxn is called a unimodular Poisson manifold. Remark 5. For the case of linear Poisson structures (Lie–Poisson brackets) the unimodular Poisson manifolds correspond to the well-known unimodular Lie algebras (see, e.g., [33]). Below we distinguish between three cases: – there is an invariant measure (without singularities) which exists on the entire manifoldMn and is defined by an explicit solution of the Liouville equation (2); – the Liouville equation (2) admits an explicit solution for density ρ(x) which has singular- ities at some points of the phase space; we shall call such a measure singular; – it is impossible to obtain an explicit Liouville solution, and in the phase space of the system there exist attracting invariant sets – attractors1 (fixed points, limit cycles, strange attractors etc.); in this case we shall say that there exists no invariant measure. Further, we illustrate the above constructions and considerations by various nonholonomic systems. Nonholonomic systems are characterized by the presence of nonintegrable constraints, resulting in fairly general systems of differential equations, which in the case of homogeneous (in velocities) constraints possess a first energy integral (see, e.g., [15] for details). The problem of existence of an invariant measure for nonholonomic systems is discussed, for example, in [15, 17, 24, 30]. In all the examples of Hamiltonizable systems considered here the dimension of the phase spaceMn is equal to five (n = 5). Moreover, depending on the existence of an invariant measure and global Casimir functions, a large number of types of Hamiltonian systems can arise (with a Poisson bracket of nonmaximal rank). Nevertheless, as a rule, it is impossible to detect all these types in practice. We list here combinations that are encountered in the examples considered. 1) There exist an invariant measure and three global Casimir functions: a nonholonomic hinge, one of the bodies is a plate (see Section 3.3); 2) There exist an invariant measure and two global Casimir functions: a nonholonomic hinge in the general case (see Section 3.4); 3) There exist an invariant measure and one global Casimir function: the Suslov problem under the condition that the constraint is imposed along the principal axis of inertia (see Section 4.2); 1As a rule, attractors can be detected only by numerical investigations. The Hojman Construction and Hamiltonization of Nonholonomic Systems 7 Figure 1. A system of two bodies coupled by a nonholonomic hinge. 4) There exist a singular invariant measure and three global Casimir functions: the Chaplygin sleigh on a horizontal plane (see Section 5.2); 5) There are no invariant measure and no global Casimir functions: the Suslov problem in a gravitational field (see Section 4.3) and the Chaplygin sleigh on an inclined plane (see Section 5.3). 3 Nonholonomic hinge 3.1 Equations of motion Consider the problem of the free motion of a system of two bodies coupled by a nonholonomic hinge. The outer body is a homogeneous spherical shell inside which a rigid body moves; the rigid body is connected with the shell by means of sharp wheels in such a way as to exclude relative rotations about the vector e fixed in the inner body (see Fig. 1). In what follows we shall call this system a nonholonomic hinge (this problem was previously studied in [3, 4]). Choose a moving coordinate system Cx1x2x3 attached to the inner body. Then, if we denote the angular velocities of the inner body and the spherical shell by ω and Ω, the constraint takes the form ω3 − Ω3 = 0. In the absence of external forces the evolution of the angular velocities is governed by the following equations: Ω̇1 = ω3(Ω2 − ω2), Ω̇2 = ω3(ω1 − Ω1), I1ω̇1 = (I2 − I3)ω2ω3, I2ω̇2 = (I3 − I1)ω1ω3, (Is + I3)ω̇3 = Is(Ω1ω2 − Ω2ω1) + (I1 − I2)ω1ω2, (11) where I = diag(I1, I2, I3) is the tensor of inertia of the inner body and Is is the tensor of inertia of the shell. 3.2 First integrals and the Poisson bracket The system (11) possesses a standard invariant measure and conserves the energy: dΩ1dΩ2dω1dω2dω3, E = 1 2 Is(Ω 2 1 + Ω2 2) + 1 2 ( I1ω 2 1 + I2ω 2 2 + (I3 + Is)ω 2 3 ) . 8 I.A. Bizyaev, A.V. Borisov and I.S. Mamaev In order to use Theorem 1, we choose u = 1 ω3 ∂ ∂ω3 as u. Then, if we denote the initial vector field (11) by v, we obtain [v,u] = −ω−13 v, u(E) = 2(I3 + Is). Thus, Theorem 1 holds and the system (11) can be represented in Hamiltonian form with a Poisson bracket in the variables x = (Ω1,Ω2, ω1, ω2, ω3) of the form J(2) = v ∧ u =  0 0 0 0 Ω2 − ω2 0 0 0 0 −(Ω1 − ω1) 0 0 0 0 I2−I3 I1 ω2 0 0 0 0 − I1−I3 I2 ω1 ∗ ∗ ∗ ∗ 0  , where the asterisks denote nonzero matrix entries resulting from the skew-symmetry condi- tion J(2). We note that even though the vector field u has a singularity when ω3 = 0, the resulting Poisson structure for the system is smooth. The found Poisson bracket corresponds to the solvable Lie algebra. According to the clas- sification of [37], this is the algebra Aspq5,17 with p = q = 0 (see [4] for details). 3.3 A flat inner body (I3 = I1 + I2) In the case where the inner body is flat, i.e., I3 = I1+I2, for J(2) we find all k = n−rank J(2) = 3 Casimir functions C1(x) = ω2 1 + ω2 2, C2(x) = Ω2 1 + (ω2 − Ω2) 2, C3(x) = Ω2 2 + (ω1 − Ω1) 2, i.e., we obtain a case that is the simplest from the point of view of integrability. Indeed, let us restrict the system (11) to the symplectic leaf by making the change of variables ω1 = c1 cosϕ, ω2 = c1 sinϕ, Ω1 = c2 cos(ϕ+ c3), Ω2 = c1 sinϕ− c2 sin(ϕ+ c3), where ϕ ∈ [0, 2π) is the angular coordinate, and c1, c2, and c3 parameterize the symplectic leaf C1(x) = √ c1, C2(x) = √ c2, C3(x) = c21 − 2c1c2 cos c3 + c22. As a result, we obtain a system with one degree of freedom (ϕ, ω3) and a canonical Poisson bracket ϕ̇ = ∂H̃ ∂ω3 , ω̇3 = −∂H̃ ∂ϕ , H̃ = 1 2 ω2 3 + c21(I1 cos2 ϕ+ (Is + I2) sin2 ϕ) 2(Is + I1 + I2) − Isc1c2 sin(ϕ+ c3) Is + Is + I2 . (12) Some of its typical trajectories are shown in Fig. 2. The Hojman Construction and Hamiltonization of Nonholonomic Systems 9 Figure 2. Typical trajectories (3.3) for I1 = 2, I2 = 3, Is = 1, c1 = 1, c2 = 2, c3 = π 3 . 3.4 The general case In the general case the Poisson bracket possesses only two global Casimir functions [3, 4] C1(x) = I1(I1 − I3)ω2 1 + I2(I2 − I3)ω2 2, C2(x) = (I1ω1 − I3Ω1) 2 + (I2ω2 − I3Ω2) 2, while a third one does not exist in the general case (it is defined only locally). Indeed, let us introduce coordinates on the symplectic leaf and assume that I3 < I2 < I1: ω1 = √ c1 I1(I1 − I3) sinϕ1, ω2 = √ c1 I2(I2 − I3) cosϕ1, Ω1 = √ I1c1 I1 − I3 sinϕ1 I3 − √ c2 sinϕ2 I3 , Ω2 = √ I2c1 I2 − I3 cosϕ1 I3 − √ c2 cosϕ2 I3 , where ϕ1, ϕ2 ∈ [0, 2π) and C1(x) = c1, C2(x) = c2. In addition, we restrict the system to the level set of the energy integral H = h by making the substitution ω3(ϕ1, ϕ2) = ± √ 2h− 2Q I23 (I3 + Is) , Q = Isc2 2 + c1 2 ( IsI1 + I23 I1 − I3 sin2 ϕ1 + IsI2 + I23 I2 − I3 cos2 ϕ1 ) − Is √ c2 (√ I1c1 I1 − I3 sinϕ1 sinϕ2 + √ I2c1 I2 − I3 cosϕ1 cosϕ2 ) . As a result, the equations of motion can be represented as ϕ̇1 = kω3(ϕ1, ϕ2), ϕ̇2 = ω3(ϕ1, ϕ2), k2 = (I1 − I3)(I2 − I3) I1I2 . (13) In the case where ω3(ϕ1, ϕ2) vanishes nowhere and k is irrational, the trajectories (13) are rectilinear orbits on a torus (see Fig. 3). Such behavior is partially due to the fact that in the general case the missing solution (5) in the chosen (local) coordinates is represented as ϕ1 − kϕ2 = const, whence it follows that the projection of the symplectic leaf on (ϕ1, ϕ2) is multi-valued. A topological analysis of the system (11) is presented in [3]. In particular, it is shown that in addition to a torus there are two other types of integral surfaces: a sphere and a sphere with three handles (two-dimensional orientable surface of genus 3). 10 I.A. Bizyaev, A.V. Borisov and I.S. Mamaev Figure 3. Trajectory (13) for I1 = 5.2, I2 = 4.3, I3 = 3.9, Is = 3.6, c1 = 10, c2 = 40, h = 10. Figure 4. Realization of the Suslov problem. 4 The Suslov problem 4.1 Equations of motion and first integrals If we assume the spherical shell from the preceding example to be fixed, we obtain the so-called Suslov problem (in Vagner’s interpretation, see Fig. 4), which describes the motion of a rigid body with a fixed point subject to the nonholonomic constraint (ω, e) = 0, where ω is the angular velocity of the body and e is the vector fixed in the body. Choose a moving coordinate system attached to the inner body, with one of the axes directed along e. Then the tensor of inertia of the moving body can be represented as I = I11 0 I13 0 I22 I23 I13 I23 I33  . In the axisymmetric potential field U = U(γ) the problem reduces to the investigation of the following closed system of equations [10]: I11ω̇1 = −ω2(I13ω1 + I23ω2) + γ2 ∂U ∂γ3 − γ3 ∂U ∂γ2 , I22ω̇2 = ω1(I13ω1 + I23ω2) + γ3 ∂U ∂γ1 − γ1 ∂U ∂γ3 , γ̇1 = −γ3ω2, γ̇2 = γ3ω1, γ̇3 = γ1ω2 − γ2ω1, (14) The Hojman Construction and Hamiltonization of Nonholonomic Systems 11 where γ = (γ1, γ2, γ3) is the unit vector of the fixed coordinate system referred to the moving axes. The system (14) conserves the energy and possesses the geometrical integral E = 1 2 ( I11ω 2 1 + I22ω 2 2 ) + U, F = γ21 + γ22 + γ23 = 1. We now consider successively several particular cases (14). 4.2 Case I13 = I23 = 0 and U = U(γ1, γ2) Suppose that I13 = I23 = 0, i.e., the constraint is imposed along the principal axis of inertia and the potential field has the form U = U(γ1, γ2). Then the system (14) possesses a standard invariant measure. Further, we introduce the vector field u = 1 γ3 ∂ ∂γ3 , for which [v,u] = −γ−13 v, u(F ) = 2. Consequently, it follows from Theorem 1 that the initial system (14) can be represented in Hamiltonian form with the Poisson bracket J(2) = v ∧ u =  0 0 0 0 − 1 I11 ∂U ∂γ2 0 0 0 0 1 I22 ∂U ∂γ1 0 0 0 0 −ω2 0 0 0 0 ω1 ∗ ∗ ∗ ∗ 0  , where the asterisks denote nonzero matrix entries resulting from the skew-symmetry condi- tion J(2). Proposition 2. In the general case, the bracket J(2) possesses one global Casimir function – the energy integral E. Proof. First of all, we note that the Casimir functions J(2) are simultaneously the integrals of the system ω̇1 = − 1 I11 ∂U ∂γ2 , ω̇2 = 1 I22 ∂U ∂γ1 , γ̇1 = −ω2, γ̇2 = ω1, (15) which is Hamiltonian with Hamiltonian E and the Poisson bracket {ω1, γ2} = − 1 I1 , {ω2, γ1} = 1 I2 . Further, if we introduce new variables p1 = √ I1ω1, p2 = √ I2ω2, q1 = √ I2γ1, q2 = √ I1γ2, then the system (15) reduces to investigating a vector field with a canonical Poisson bracket and a Hamiltonian of the form H = 1 2 (p21 + p22) + U ( q1 I2 , q2 I1 ) . (16) This is a natural system which describes the motion of a material point on a plane and in which, as is well known, there are generally no additional first integrals, and hence the resulting Poisson bracket has no remaining global Casimir functions. � 12 I.A. Bizyaev, A.V. Borisov and I.S. Mamaev Remark 6. However, the system (16) has the well-know potentials U for which there exist (one or two) additional integrals (and hence Casimir functions) of different degrees in momenta (see, e.g., the recent papers [34, 35] or [26, 27] for the Hénon–Heiles system). 4.3 Case I13 6= 0, I23 6= 0 and U = (a, γ) Now consider the more general situation I13 6= 0, I23 6= 0, but in the potential field U = (a,γ). In this case, the equations of motion are invariant under the transformation ω1 → λω1, ω2 → λω2, γ → λ2γ, dt→ λ−1dt, which is associated to the vector field û = ω1 ∂ ∂ω1 + ω2 ∂ ∂ω2 + 2γ1 ∂ ∂ω1 + 2γ2 ∂ ∂ω2 + 2γ3 ∂ ∂ω3 , for which we have [u,v] = v, u(E) = 2E. Thus, the system under consideration can be represented in conformally Hamiltonian form with the Poisson bracket J(2) = v ∧ u. In the case at hand, J(2) has a rather cumbersome form, so we do not present it here explicitly. We note that it is cubic in the velocities ω. Proposition 3. In the general case, the system (14) has no invariant measure with smooth density in the potential field U = (a,γ). Proof. In the potential field U = (a,γ) the system (14) possesses a fixed point ω1 = 0, ω2 = − a3 I23 , γ1 = 0, γ2 = 1, γ3 = 0 having a nonzero trace of the linearization matrix: I13 I1 √ a3√ I23 . Consequently, the system considered has no invariant measure with smooth density [30, 31, 32] in a neighborhood of the fixed point. Indeed, in this case the characteristic polynomial of the linearized system is represented as P (λ) = λ2 ( λ3 − c1λ2 + c2λ+ c3 ) , c1 = I13 √ a3 I1 √ I23 , c2 = a2 I1 − I1I2 + 2I223 I23I1I2 a3, c3 = √ a3I23 ( I13a3 I1I223 − 2a1 I1I2 ) . This polynomial is not reciprocal. � In this system, an analogous result was obtained for U = 0 previously in [30]. Later, however, a singular invariant measure [10] was found in this case. We also note that the pair of brackets that was constructed for the Suslov problem is defined in the extended phase space, as are the brackets described above. R2 × R3 = {(ω1, ω2, γ1, γ2, γ3)}. The Hojman Construction and Hamiltonization of Nonholonomic Systems 13 Indeed, for these brackets the geometric integral F = γ21 + γ22 + γ23 is not a Casimir function, so it is not clear how to restrict them to the actual phase space of the system R2 × S2 = {(ω1, ω2, γ1, γ2, γ3), γ 2 1 + γ22 + γ23 = 1}. In view of the above discussion (at the end of Section 3.2), these brackets cannot provide any insight into the system’s properties. Remark 7. It may seem that a singular invariant measure with density having singularities in the considered family of fixed points is possible in this case too (see, e.g., Section 5). However, it is impossible to obtain a suitable solution of the Liouville equation in this case. Moreover, as numerical investigations show, this system may also have more complicated attracting sets – limit cycles. Numerical investigations also show that this system has no additional real-analytical integrals. Remark 8. A proof of the absence of meromorphic integrals for other (more particular) cases of the Suslov problem was obtained in [25, 36, 41]. The methods developed in these papers can be used in this case also, since the system (14) has a particular periodic solution which for I13 = 0, a1 = 0, and a2 = 0 has the form ω1(t) = 2k cn(t, k), ω2(t) = 0, γ1(t) = 0, γ2(t) = −2I1k a3 sn(t, k) dn(t, k), γ3(t) = I1 a3 ( 2k2 sn2(t, k)− 1 ) , H = I1 ( 2k2 − 1 ) , where sn(t, k), cn(t, k), dn(t, k) are the elliptic Jacobi functions with parameter k. Remark 9. We note that in [10, 32] other cases of representation of the equations of the Suslov problem were found in Hamiltonian form, with a Poisson bracket of rank 4, which cannot be obtained with the help of the Hojman construction. 5 The Chaplygin sleigh 5.1 Equations of motion The Chaplygin sleigh is a rigid body moving on a horizontal plane and supported at three points: two (absolutely) smooth posts and a knife edge such that the body cannot move perpendicularly to the plane of the wheel. As a historical remark, we note that although the Chaplygin sleigh is usually linked to the works of Chaplygin [21] and Caratheodory [19], they were considered slightly earlier by Brill [40] as an example of the mechanism of a nonholonomic planimeter. In this case, the configuration space coincides with the group of motions of the plane SE(2). To parameterize it, we choose Cartesian coordinates (x, y) of point O1 (see Fig. 5) in the coor- dinate system Oxy and the angle ϕ of rotation of the axes O1x1x2 relative to Oxy. The motion of the Chaplygin sleigh in the variables (v1, ω, x, y, ϕ) is described by the equa- tions (see [13] for details) mv̇1 = maω2 − ∂U ∂x cosϕ− ∂U ∂y sinϕ, (I +ma2)ω̇ = −maωv1 − ∂U ∂ϕ , ẋ = v1 cosϕ, ẏ = v1 sinϕ, ϕ̇ = ω, (17) where v1 is the projection of the velocity of point O1 on the axis O1x1, and m and I are, respectively, the mass and the moment of inertia of the body, a is the distance specifying the position of the knife edge, and U is the potential of the external forces. 14 I.A. Bizyaev, A.V. Borisov and I.S. Mamaev Figure 5. The Chaplygin sleigh on a plane. The system (17) conserves the energy integral E = 1 2 ( mv21 + ( I +ma2 ) ω2 ) + U. Below we consider successively several particular cases. 5.2 The Chaplygin sleigh on a horizontal plane Symmetry fields and invariant measure. In the absence of an external potential field U = 0 the system (17) admits the symmetry group SE(2) which is associated to the vector fields ux = ∂ ∂x , uy = ∂ ∂y , uϕ = ∂ ∂ϕ , which are symmetry fields. Moreover, (17) possess a singular invariant measure ω−1dv1dωdxdydϕ. Poisson bracket. We note that the system (17) is invariant under the transformation v1 → λv1, ω → λω, dt→ λ−1dt, which is associated to the vector field u = v1 ∂ ∂v1 + ω ∂ ∂ω , and [u,v] = v, u(E) = 2E, where v is the initial vector field defined by the system (17). Consequently, in the case E 6= 0 the system (17) is represented in Hamiltonian form ẋ = J(2)∂E ∂x , x = (v1, ω, x, y, ϕ), The Hojman Construction and Hamiltonization of Nonholonomic Systems 15 with the Poisson bracket J(2) = v ∧ u 2E =  0 aω I+ma2 − v21 cosϕ mv21+(I+ma2)ω2 − v21 sinϕ mv21+(I+ma2)ω2 − ωv1 mv21+(I+ma2)ω2 ∗ 0 − v1ω cosϕ mv21+(I+ma2)ω2 − v1ω sinϕ mv21+(I+ma2)ω2 − ω2 mv21+(I+ma2)ω2 ∗ ∗ 0 0 0 ∗ ∗ ∗ 0 0 ∗ ∗ ∗ ∗ 0  , where the asterisks denote the matrix entries resulting from the skew-symmetry condition J(2). We examine in more detail a symplectic foliation which is defined by the bracket J(2). First of all, we note that the manifold M4 0 = {(ω, v1, x, y, ϕ) \|ω = 0} defines a Poisson submanifold with the Casimir functions C1 = ϕ, C2 = x sinϕ− y cosϕ. Thus, the entire phase space of the system M5 is a union of three nonintersecting Poisson submanifolds M5 =M5 + ∪M4 0 ∪M5 −, M5 + = {(ω, v1, x, y, ϕ) \|ω > 0}, M5 − = {(ω, v1, x, y, ϕ) \|ω < 0}. Let us consider M5 + in more detail and pass from (v1, ω) to (polar) coordinates (h, ψ): v1 = Aah cosψ, ω = h sinψ, A2 = 1 + I ma2 > 1, E = 1 2 mA2a2h2, where h ∈ (0,∞) and ψ ∈ (0, π). In the new variables (ψ, x, y, ϕ, h) the Poisson bracket becomes J(2) =  0 0 0 0 −hA−1 sinψ 0 0 0 0 hAa cosψ cosϕ 0 0 0 0 hAa cosψ sinϕ 0 0 0 0 h sinψ ∗ ∗ ∗ ∗ 0  . (18) As a result, the Casimir functions (18) have the form C1 = Aψ + ϕ, C2 = x+A2a ∫ ψ ψ∗ tan(ξ) cos(Aξ − C1)dξ, C3 = y −A2a ∫ ψ ψ∗ tan(ξ) sin(Aξ − C1)dξ, (19) where ψ∗ ∈ (0, π). The level surface of the Casimir function C1 = const foliates the domain M5 + into four- dimensional surfaces. The projection of each such surface into the three-dimensional space S1 × R2 + = {(ϕ, ω, v1) \|ω > 0} is a ruled surface (similar to a helicoid). Its equation can be represented in the following parametric form v1 = Aah cos ( ϕ+ C1 A ) , ω = h sin ( ϕ+ C1 A ) , ϕ = ϕ. (20) 16 I.A. Bizyaev, A.V. Borisov and I.S. Mamaev Figure 6. Characteristic surface defined by the Casimir function C1 in the domainM5 + for C1 = 2 and A = 3. In the definition (20) the periodicity in ϕ should, of course, be taken into account (see Fig. 6). Proceeding in a similar way for M5 −, it is easy to check that relations (19) also define the Casimir functions with the only difference that ψ∗ ∈ (π, 2π). The second Poisson bracket. In this case it turns out that when E 6= 0, by Proposition 1, the system (17) can be represented in conformally Hamiltonian form ẋ = (2E)(−1)J(4)∂E ∂x , with a Poisson bracket of rank 4 using the symmetry fields as follows: ûx and ûy J(4) = v ∧ u+ ux ∧ uy =  0 aω3 + ma I+ma2 ωv21 −v21 cosϕ −v21 sinϕ −ωv1 ∗ 0 −v1ω cosϕ −v1ω sinϕ −ω2 ∗ ∗ 0 1 0 ∗ ∗ ∗ 0 0 ∗ ∗ ∗ ∗ 0  . Remark 10. We note that the Jacobi identity for the matrix J(4) E does not hold. In this case, ω = 0 defines the Poisson submanifold J(4) on which rank J(4) = 2, and the Casimir functions are C1 = ϕ, C2 = Aa(x sinϕ− y cosϕ)− h−1. In the remaining region of the phase space J possesses the only Casimir function C1 = Aψ + ϕ, tanϕ = v2 Aaω . The bracket J(4) was obtained previously in the initial coordinates (v1, ω, x, y, ϕ) in [13]. 5.3 The Chaplygin sleigh on an inclined plane Below we consider the motion of the Chaplygin sleigh on an inclined plane. We direct the axis Ox along the line of maximum slope, then U = mµ(x+ a cosϕ), µ = g sinχ, The Hojman Construction and Hamiltonization of Nonholonomic Systems 17 where χ is the angle of inclination of the plane to the horizon. In this case, the energy integral has the form E = I +ma2 2 ω2 + mv21 2 +mµ(x+ a cosϕ). Moreover, two symmetry fields are preserved in the system (17) ux = ∂ ∂x , uy = ∂ ∂y . Since ux(E) = mµ, Theorem 1 applies to this system. As a result, its equations of motion are represented in Hamiltonian form with the Poisson bracket J(2) of rank 2: J(2) = v ∧ ux =  0 ω 0 0 0 ∗ 0 −v1 sinϕ µ cosϕ− aω2 ma I+ma2 (ωv1 − µ sinϕ) ∗ ∗ 0 0 0 ∗ ∗ ∗ 0 0 ∗ ∗ ∗ ∗ 0  , where v is the initial vector field defined by the system (17). In [13] it is shown that the dynamics of this system, which is noncompact, has a complicated, seemingly random behavior, leading to the absence of an invariant measure and global analytic Casimir functions in this system. 6 Conclusion Thus, in this paper it has been shown that the Hojman construction (using conformal symmetry fields) allows one to obtain Poisson brackets for various nonholonomic systems. As a rule, the resulting Poisson brackets do not possess a maximal rank, and in the general case a smooth invariant measure and global Casimir functions may be absent for these brackets. 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Surv. 52 (1997), 434–435. http://dx.doi.org/10.1007/BF01077727 http://dx.doi.org/10.1070/RD2002v007n02ABEH000203 http://arxiv.org/abs/nlin.SI/0503027 http://dx.doi.org/10.1016/j.physleta.2008.06.065 http://dx.doi.org/10.1088/1751-8113/43/38/382001 http://arxiv.org/abs/1004.3854 http://dx.doi.org/10.1063/1.4763464 http://dx.doi.org/10.1063/1.522992 http://dx.doi.org/10.1063/1.522992 http://dx.doi.org/10.1134/S1560354712010078 http://dx.doi.org/10.1134/S1560354712010078 http://arxiv.org/abs/1106.1952 http://dx.doi.org/10.1134/S1560354712050061 http://dx.doi.org/10.1134/S1560354712050061 http://dx.doi.org/10.1070/RM1997v052n02ABEH001805 1 Introduction 2 Tensor invariants and Hamiltonization of dynamical systems 2.1 Basic notions and definitions 2.2 The general Hojman construction 2.3 Invariant measure in Hamiltonian systems 3 Nonholonomic hinge 3.1 Equations of motion 3.2 First integrals and the Poisson bracket 3.3 A flat inner body (I3=I1+I2) 3.4 The general case 4 The Suslov problem 4.1 Equations of motion and first integrals 4.2 Case I13=I23=0 and U=U(1, 2) 4.3 Case I13=0, I23=0 and U=(a, ) 5 The Chaplygin sleigh 5.1 Equations of motion 5.2 The Chaplygin sleigh on a horizontal plane 5.3 The Chaplygin sleigh on an inclined plane 6 Conclusion References

The Hojman Construction and Hamiltonization of Nonholonomic Systems

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