Integrability of Nonholonomic Heisenberg Type Systems

We show that some modern geometric methods of Hamiltonian dynamics can be directly applied to the nonholonomic Heisenberg type systems. As an example we present characteristic Killing tensors, compatible Poisson brackets, Lax matrices and classical r-matrices for the conformally Hamiltonian vector f...

Ausführliche Beschreibung

Gespeichert in:

Bibliographische Detailangaben
Veröffentlicht in:	Symmetry, Integrability and Geometry: Methods and Applications
Datum:	2016
Hauptverfasser:	Grigoryev, Y. A., Sozonov, A.P., Tsiganov, A.V.
Format:	Artikel
Sprache:	English
Veröffentlicht:	Інститут математики НАН України 2016
Online Zugang:	https://nasplib.isofts.kiev.ua/handle/123456789/148541
Tags:	Tag hinzufügen Keine Tags, Fügen Sie den ersten Tag hinzu!
Назва журналу:	Digital Library of Periodicals of National Academy of Sciences of Ukraine
Zitieren:	Integrability of Nonholonomic Heisenberg Type Systems / Y. A. Grigoryev, A.P. Sozonov, A.V. Tsiganov // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 25 назв. — англ.

Institution

Digital Library of Periodicals of National Academy of Sciences of Ukraine

id	nasplib_isofts_kiev_ua-123456789-148541
record_format	dspace
spelling	Grigoryev, Y. A. Sozonov, A.P. Tsiganov, A.V. 2019-02-18T14:53:57Z 2019-02-18T14:53:57Z 2016 Integrability of Nonholonomic Heisenberg Type Systems / Y. A. Grigoryev, A.P. Sozonov, A.V. Tsiganov // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 25 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 37J60; 70G45; 70H45 DOI:10.3842/SIGMA.2016.112 https://nasplib.isofts.kiev.ua/handle/123456789/148541 We show that some modern geometric methods of Hamiltonian dynamics can be directly applied to the nonholonomic Heisenberg type systems. As an example we present characteristic Killing tensors, compatible Poisson brackets, Lax matrices and classical r-matrices for the conformally Hamiltonian vector fields obtained in a process of reduction of Hamiltonian vector fields by a nonholonomic constraint associated with the Heisenberg system. We are very grateful to the referees for thorough analysis of the manuscript, constructive suggestions and proposed corrections, which certainly lead to a more profound discussion of the results. We are also deeply grateful A.V. Borisov and I.A. Bizayev for the relevant discussion. Section 2 was written by A.V. Tsiganov and supported by the Russian Science Foundation (project 15-12-20035). Section 3 was written by Yu.A. Grigoryev and A.P. Sozonov within the framework of the Russian Science Foundation (project 15-11-30007). en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Integrability of Nonholonomic Heisenberg Type Systems Article published earlier
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
title	Integrability of Nonholonomic Heisenberg Type Systems
spellingShingle	Integrability of Nonholonomic Heisenberg Type Systems Grigoryev, Y. A. Sozonov, A.P. Tsiganov, A.V.
title_short	Integrability of Nonholonomic Heisenberg Type Systems
title_full	Integrability of Nonholonomic Heisenberg Type Systems
title_fullStr	Integrability of Nonholonomic Heisenberg Type Systems
title_full_unstemmed	Integrability of Nonholonomic Heisenberg Type Systems
title_sort	integrability of nonholonomic heisenberg type systems
author	Grigoryev, Y. A. Sozonov, A.P. Tsiganov, A.V.
author_facet	Grigoryev, Y. A. Sozonov, A.P. Tsiganov, A.V.
publishDate	2016
language	English
container_title	Symmetry, Integrability and Geometry: Methods and Applications
publisher	Інститут математики НАН України
format	Article
description	We show that some modern geometric methods of Hamiltonian dynamics can be directly applied to the nonholonomic Heisenberg type systems. As an example we present characteristic Killing tensors, compatible Poisson brackets, Lax matrices and classical r-matrices for the conformally Hamiltonian vector fields obtained in a process of reduction of Hamiltonian vector fields by a nonholonomic constraint associated with the Heisenberg system.
issn	1815-0659
url	https://nasplib.isofts.kiev.ua/handle/123456789/148541
citation_txt	Integrability of Nonholonomic Heisenberg Type Systems / Y. A. Grigoryev, A.P. Sozonov, A.V. Tsiganov // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 25 назв. — англ.
work_keys_str_mv	AT grigoryevya integrabilityofnonholonomicheisenbergtypesystems AT sozonovap integrabilityofnonholonomicheisenbergtypesystems AT tsiganovav integrabilityofnonholonomicheisenbergtypesystems
first_indexed	2025-11-24T09:17:30Z
last_indexed	2025-11-24T09:17:30Z
_version_	1850454951279460352
fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 12 (2016), 112, 14 pages Integrability of Nonholonomic Heisenberg Type Systems Yury A. GRIGORYEV †, Alexey P. SOZONOV † and Andrey V. TSIGANOV †‡ † St. Petersburg State University, St. Petersburg, Russia E-mail: yury.grigoryev@gmail.com, sozonov.alexey@yandex.ru, andrey.tsiganov@gmail.com ‡ Udmurt State University, Izhevsk, Russia Received March 17, 2016, in final form November 22, 2016; Published online November 25, 2016 http://dx.doi.org/10.3842/SIGMA.2016.112 Abstract. We show that some modern geometric methods of Hamiltonian dynamics can be directly applied to the nonholonomic Heisenberg type systems. As an example we present characteristic Killing tensors, compatible Poisson brackets, Lax matrices and classical r- matrices for the conformally Hamiltonian vector fields obtained in a process of reduction of Hamiltonian vector fields by a nonholonomic constraint associated with the Heisenberg system. Key words: Hamiltonian dynamics; nonholonomic systems 2010 Mathematics Subject Classification: 37J60; 70G45; 70H45 1 Introduction Hamiltonian mechanics emerged in 1833 as a convenient reformulation of classical Newtonian mechanics and analytical Lagrangian mechanics. Over the next almost two centuries, it was gradually realized that Hamiltonian formulation of a system of differential equations d dt xk = Xk, X = PdH has several advantages based on the main postulate that evolution of a physical system over time is governed by a single Hamiltonian H of that system and a Poisson bivector P describing geometry and topology of the phase space. Many different mathematical methods and concepts are widely used in the Hamiltonian formalism. For instance, we can construct first integrals of the Hamiltonian vector field X using Killing tensors (Riemannian geometry), compatible Poisson brackets (bi-Hamiltonian geometry) or Lax matrices (algebraic geometry, Lie algebras theory, classical r-matrix theory etc.). It is natural to ask what happens to these mathematical methods when we impose nonholo- nomic constraints on the Hamiltonian system and make a suitable reduction. Usually presence of the constraints drastically modifies or destroys these geometric constructions even when one gets a conformally Hamiltonian vector field X̂ = µ(x)P̂ dĤ after the reduction, see [2, 5, 7, 8, 9, 10, 11, 15, 22, 24] and references within. Here Ĥ and P̂ are Hamiltonian and Poisson bivector on reduced phase space, whereas function µ(x) is the so-called conformal factor. The main aim of this note is to present a family of conformally Hamiltonian dynamical systems for which all the geometric methods listed above can be applied without any addi- tional modifications. The corresponding nonholonomic constraint is associated with the so- called Heisenberg system or nonholonomic integrator, which plays an important role in both mailto:yury.grigoryev@gmail.com mailto:sozonov.alexey@yandex.ru mailto:andrey.tsiganov@gmail.com http://dx.doi.org/10.3842/SIGMA.2016.112 2 Yu.A. Grigoryev, A.P. Sozonov and A.V. Tsiganov nonlinear control and nonholonomic dynamics [7]. We have to underline that the integrable systems discussed in this note look quite artificial from the viewpoint of control theory and mechanics but they provide an example of the standard geometric methods applicability to the conformally Hamiltonian systems. This paper is organized as follows. Section 1.1 recalls a brief description of the constrained motion in three-dimensional Euclidean space. Section 2 contains the main results on reduced motion of the nonholonomic Heisenberg type systems on the plane. We will show that integrable potentials for this non Hamiltonian vector field satisfy to the Bertrand–Darboux type equation. Solutions of this equation and the corresponding characteristic coordinates will be explicitly determined. Section 3 deals with application of the standard Stäckel theory to the conformally Hamiltonian vector field. We will discuss a construction of the Stäckel matrices, compatible Poisson structures, Lax matrices and classical r-matrices. 1.1 Main definitions According to [7, 18, 24] we take the standard Hamiltonian equations of motion in Euclidean space R3 q̇i = ∂H ∂pi , ṗi = −∂H ∂qi , i = 1, 2, 3, (1.1) where H = 1 2 ( p21 + p22 + p23 ) + V (q1, q2, q3). (1.2) On the phase space M = T ∗R3 we can introduce coordinates x = (q, p) in which equations (1.1) have form ẋi = Xi and to determine Hamiltonian vector field X = 6∑ i=1 Xi ∂ ∂xi , which is a linear operator on a space of the smooth functions on M that encodes the evolution of any quantity Ḟ = X(F ) = 6∑ i=1 Xi ∂F ∂xi . Let us impose the constraint of first order in momenta (velocities) f = (b, p) = 0, (1.3) where b = (b1, b2, b3) is a vector depending on coordinates q and (x, y) means an inner product in R3. In this case equations of motion are written in the following form q̇i = ∂H ∂pi , ṗi = −∂H ∂qi + λbi, i = 1, 2, 3, (1.4) together with the constraint equation (1.3). The corresponding vector field looks like an additive perturbation of the initial Hamiltonian vector field X̂ = X ∣∣ b=0 + λ ( b1 ∂ ∂p1 + b2 ∂ ∂p2 + b3 ∂ ∂p3 ) , Integrability of Nonholonomic Heisenberg Type Systems 3 where unknown Lagrange multiplier λ has to be computed from the condition ḟ = X̂(f) = X(f) ∣∣ b=0 + λ(b, b) = 0, such that λ = X(f)\|b=0 (b, b) . (1.5) Here X(f)\|b=0 denotes the vector field in the absence of constraint. Such equations and an equivalence of Hamiltonian and Lagrangian reductions are carefully discussed in the book [7]. For completeness and self-sufficiency of presentation we consider integrable constraint f = p3 = 0, b = (0, 0, 1), (1.6) when a third component of momenta is equal to zero, and non integrable constraint f = p3 − (q2p1 − q1p2) = 0, b = (−q2, q1, 1), (1.7) when third components of momenta and angular momenta coincide with each other. If we assume that the potential V in R3 does not depend on q3, then Lagrange multipliers (1.5) are equal to λ = 0 and λ = q1∂2V − q2∂1V 1 + q21 + q22 , ∂j = ∂ ∂qj , respectively. Thus, the Hamiltonian system (1.2) with integrable constraint (1.6) represent a Hamiltonian system on the plane R2 embedded in a three dimensional Euclidean space R3 in which there is no force acting on the third component. If we impose non integrable con- straint (1.7), the equations for q3 also decouples from the rest of the system (1.4) and we obtain a two-degrees of freedom non-Hamiltonian system on the plane. Following to [7, 18] we will call dynamical system associated with X̂ as a Heisenberg type system. Integrability of the Hamiltonian systems on the plane is the well-known problem of classical mechanics considered by Bertrand [4] and Darboux [12]. Later on results of this investigation due to Darboux were included in the textbook of Whittaker on analytical mechanics [25] almost verbatim. In the next Section we compare classical Bertrand–Darboux theorem with its non- Hamiltonian counterpart, which appears in the nonholonomic case. 2 Reduced systems on the plane When V = V (q1, q2), substituting integrable constraint p3 = 0 (1.6) into (1.2) one gets H1 = 1 2 ( p21 + p22 ) + V (q1, q2) = 1 2 gijpipj + V (q1, q2), g = ( 1 0 0 1 ) (2.1) with the summation convention in force; we regard the gij as components of the covariant form of the metric tensor g on the plane, and use this metric freely to raise and lower indices. After standard Hamiltonian reduction by cyclic third coordinate, the original Hamiltonian vector field (1.1) in T ∗R3 becomes the Hamiltonian vector field in T ∗R2 ẋ = X, X = PdH1, P = ( 0 I −I 0 ) , (2.2) 4 Yu.A. Grigoryev, A.P. Sozonov and A.V. Tsiganov where x = (q1, q2, p1, p2). The Poisson bivector P in (2.2) defines canonical Poisson bracket {qi, pi} = 1, {qi, pk} = {p1, p2} = 0, i 6= k. (2.3) Of course, this vector field X preserves Poisson structure P , energy H1 and the standard volume 2-form Ω = dq ∧ dp. Substituting non integrable constraint p3 = q2p1 − q1p2 (1.7) into (1.2) one gets Ĥ1 = 1 2 ( p21 + p22 + (q2p1 − q1p2)2 ) + V (q1, q2) = 1 2 ĝijpipj + V (q1, q2), (2.4) ĝ = ( 1 + q22 −q1q2 −q1q2 1 + q21 ) . The second order contravariant symmetric tensor field ĝ is a Killing tensor with respect to the standard metric tensor g, i.e., it satisfies to the Killing equation [[g, ĝ]] = 0, where [[·, ·]] is a Schouten bracket. It allows us to construct conformal Killing tensor of second order G = ĝ − tr (ĝ) g = − ( 1 + q21 q1q2 q1q2 1 + q22 ) with vanishing Nijenhuis torsion and to define the standard Turiel deformation of the standard symplectic form ω̂ = d ( Gijpidqj ) , see [21, 22] and references within. The corresponding Poisson bracket {qi, pi}∧ = 1 + q2i , {qi, pj}∧ = q1q2, {p1, p2}∧ = q1p2 − q2p1, i 6= j (2.5) is compatible with canonical bracket (2.3). Because the Turiel deformation is trivial deformation in the Lichnerowicz–Poisson sense, there is a change of variables p1 → π1 = q1q2p2 − ( 1 + q22 ) p1 1 + x21 + x22 , p2 → π2 = q1q2p1 − ( 1 + q21 ) p2 1 + x21 + x22 , (2.6) which transforms this bracket to the canonical one (2.3), similar to other nonholonomic systems with constraints of first order in momenta [6, 10, 20]. After the nonholonomic reduction, original Hamiltonian vector field (1.1) in T ∗R3 becomes a conformally Hamiltonian vector field in T ∗R2 ẋ = X̂, X̂ = µP̂dĤ1, P̂ = ω̂−1, (2.7) where conformal factor µ = ( 1 + q21 + q22 )−1 is a nowhere vanishing smooth function on the plane. This vector field possesses energy Ĥ1 and the volume 2-form Ω̂ = µdq ∧ dp, but it does not preserve the Poisson bivector P̂ . Integrability of Nonholonomic Heisenberg Type Systems 5 2.1 Bertrand–Darboux type equation An existence of the integrals of second order in velocities for Hamiltonian vector field (2.2) is described by a classical Bertrand–Darboux theorem [4, 12, 25]. Proposition 1. The function (2.1) H1 = 1 2 gijpipj + V (q1, q2), g = ( 1 0 0 1 ) (2.8) defines integrable Hamiltonian vector field X (2.2), which has an independent quadratic first integral H2 = Kijpipj + U(q1, q2), (2.9) if the second order contravariant symmetric tensor field K obeys the Killing tensor equation [[g,K]] = 0, (2.10) and potential V satisfies the compatibility condition d(KdV ) = 0, K = Kg−1, (2.11) where K is the tensor field of (1, 1) type. A characteristic coordinate system for (2.11) provides separation for the potential V and can be taken as one of the following four orthogonal coordinate systems on the plane: elliptic, parabolic, polar or Cartesian. If a non-trivial solution exists, the Bertrand–Darboux equation (2.11) can be reduced to canonical form by transforming to characteristic coordinates, which appear to be separation co- ordinates for the Hamilton–Jacobi equation related to the natural Hamiltonian (see Bertrand [4], Darboux [12], or Whittaker [25] and Ankiewicz and Pask [1] for a full proof of the BD theorem). The modern proof of this classic statement may be found in [19]. In nonholonomic case we can also substitute the same ansatz (2.9) into the equation Ḣ2 = 0 and get the following generalisation of the Bertrand–Darboux result. Proposition 2. The function (2.4) Ĥ1 = 1 2 ĝijpipj + V (q1, q2), ĝ = ( 1 + q22 −q1q2 −q1q2 1 + q21 ) (2.12) defines integrable conformally Hamiltonian vector field X̂ (2.7), which has an independent quad- ratic first integral Ĥ2 = Kijpipj + U(q1, q2), (2.13) if the second order contravariant symmetric tensor field K is a Killing tensor [[g,K]] = 0, g = ( 1 0 0 1 ) (2.14) and potential V satisfies the compatibility condition d ( K̂dV ) = 0, K̂ = Kĝ−1. (2.15) A characteristic coordinate system for (2.15) provides separation for potential V (q1, q2). 6 Yu.A. Grigoryev, A.P. Sozonov and A.V. Tsiganov In this case existence of the second integral of motion guarantees integrability of the given non Hamiltonian vector field X̂ according to the Euler–Jacobi theorem [17]. Equation (2.15) was briefly discussed in [23] using coordinates q1,2 and momenta π1,2 (2.6). Remark 1. In contrast with the standard Bertrand–Darboux theorem the Killing equa- tion (2.14) and compatibility condition (2.15) include two different metrics g and ĝ. It happens because the reaction force is entered in the second part of the vector field (1.4) only. Sequentially, K is a Killing tensor with respect to the Euclidean metric g to the plane, which is an induced metric associated with standard embedding R2 ⊂ R3 in Cartesian coordinates. However, K is not a Killing tensor with respect to metric ĝ, which appears in the reduced Hamiltonian (2.4) after nonholonomic reduction and then defines raising and lowering indices in the compatibility condition (2.15). 2.2 Solutions of the Bertrand–Darboux type equation In [12] Darboux proceeds to find the unknown potential V (q1, q2) by solving the compatibil- ity condition (2.11) using method of characteristics. Solving the second compatibility condi- tion (2.15) we can apply the same method. In both cases characteristic coordinates for (2.11) and (2.15) consist of eigenvalues of the corresponding (1, 1) tensor field K or K̂. In order to describe these coordinates we can start with a well-known generic solution of the common Killing tensor equations (2.10) and (2.14) K = ( c1q 2 2 + 2c2q2 + c3 −c1q1q2 − c2q1 − c4q2 + c6 −c1q1q2 − c2q1 − c4q2 + c6 c1q 2 1 + 2c4q1 + c5 ) , (2.16) which depends on six constants of integration c1, . . . , c6. Substituting this generic solution into the standard compatibility condition (2.11) one gets( A∂11 +B∂22 + C∂12 + a∂1 + b∂2 ) V (q1, q2) = 0, (2.17) where ∂i = ∂/∂qi, ∂ik = ∂2/∂qi∂qk and polynomials of second order read as A = c1q1q2 + c2q1 + c4q2 − c6, B = −A, C = c1q 2 2 − c1q21 + 2c2q2 − 2c4q1 + c3 − c5, whereas polynomials of the first order are a = 3(c1q2 + c2), b = −3(c1q1 + c4). The linear PDE (2.17) of second order was obtained by Bertrand [4] and studied by Darboux [12]. Thus, it was later called the Bertrand–Darboux equation [19, 25]. Solving (2.17) for V (q1, q2) amounts to finding admissible potentials of the Hamiltonian systems defined by H (2.1), which integrability is afforded by the existence of first integrals (2.9) which are quadratic in the momenta. Second potential U(q1, q2) in (2.9) is a solution of the equation d(KdU) = V. In nonholonomic case, substituting the same generic solution (2.16) into the compatibility condition (2.15) one gets similar linear PDE( 1 + q21 + q22 )( Â∂11 + B̂∂22 + Ĉ∂12 ) V (q1, q2) + ( â∂1 + b̂∂2 ) V (q1, q2) = 0, (2.18) with coefficients Â = ( 1 + q21 ) (q1c2 − c6) + q2 ( 1− q21 ) c4 + q1q2(c1 − c5), Integrability of Nonholonomic Heisenberg Type Systems 7 B̂ = − ( 1 + q22 ) (q2c4 − c6)− q1 ( 1− q22 ) c2 − q1q2(c1 − c3), Ĉ = − ( q21 − q22 ) c1 + ( 1 + q21 ) (2q2c2 + c3)− ( 1 + q22 ) (2q1c4 + c5), and â = ( q21 + q22 + 3 )( c1q2 + 2c2q 2 1 − 2c4q1q2 ) − q2 ( 2 ( q21 + 1 ) c3 − ( q21 − q22 − 1 ) c5 ) + q1 ( q21 − 3q22 + 1 ) c6 − ( q21 + q22 − 3 ) c2, b̂ = − ( q21 + q22 + 3 )( c1q1 + 2c4q 2 2 − 2c2q1q2 ) + q1 ( 2 ( q22 + 1 ) c5 + ( q21 − q22 + 1 ) c3 ) + q2 ( 3q21 − q22 − 1 ) c6 + ( q21 + q22 − 3 ) c4. Solutions V (q1q2) of this Bertrand–Darboux type equation (2.18) determine all the admissible Jacobi integrals (2.4), which define integrable conformally Hamiltonian vector fields (2.7) with first integrals (2.13) of second order in momenta. In this case potential U(q1, q2) in (2.13) is a solution of the equation d ( K̂dU ) = V. Remark 2. Before solving the PDE (2.17), Darboux ingeniously observes that it can be sim- plified without loss of generality [12]. Indeed, by rotating and translating the axes, one can simplify the general solution of the Killing tensor equation, thus bringing it to a certain canoni- cal form. In modern language solving the equivalence and canonical forms problem for the Killing tensor equations (2.10), (2.14) is equivalent to analysing the orbits of the six-dimensional vector space of solutions under the action of the Lie group of orientation-preserving isometries. It is well known that there are four types of orbits generated by the following canonical Killing tensors: Cartesian: K(1) = ( 1 0 0 0 ) , polar: K(2) = ( q22 −q1q2 −q1q2 q21 ) , parabolic: K(3) = ( 0 −q2 −q2 2q1 ) , elliptic: K(4) = ( κ2 + q22 −q1q2 −q1q2 q21 ) , κ ∈ R, (2.19) associated with Cartesian, polar, parabolic and elliptic coordinate systems on the plane, respec- tively. Let us consider these characteristic coordinate systems associated with the Killing ten- sors (2.19). Elliptic coordinates. Substituting Killing tensor K(4) into the (1,1) tensor K = K(4)g−1 and calculating its eigenvalues one gets elliptic coordinates ξ1,2 = κ+ q21 + q22 ± √ κ2 − 2 ( q21 − q22 ) κ+ ( q21 + q22 )2 up to the constant factor. In nonholonomic case, substituting the same tensor K(4) into the characteristic tensor K̂ = K(4)ĝ−1, one gets the following eigenvalues ξ̂1,2 = ( 1 + q21 ) κ+ q21 + q22 ± √( 1 + q21 )2 κ2 − 2 ( q21 ( q21 + q22 ) + q21 − q22 ) κ+ ( q21 + q22 )2 1 + q21 + q22 . 8 Yu.A. Grigoryev, A.P. Sozonov and A.V. Tsiganov Solutions of the compatibility equations (2.11) and (2.15) are labelled by two arbitrary func- tions F1 and F2 on these coordinates: V (q1, q2) = F1(u1)− F2(u2) u1 − u2 , u1,2 = ξ1,2 or u1,2 = ξ̂1,2. Parabolic coordinates. In Hamiltonian case eigenvalues of the characteristic tensor K = K(3)g−1 are standard parabolic coordinates ζ1,2 = q1 ± √ q21 + q22, (2.20) whereas in nonholonomic case eigenvalues of the characteristic tensor K̂ = K(3)ĝ−1 read as ζ̂1,2 = q1 ± √ q22 + 1 √ q21 + q22 1 + q21 + q22 . (2.21) It is easy to express new characteristic coordinates via standard ones ζ̂1,2 = 2 ( ζ1 + ζ2 ± (ζ1 − ζ2) √ 1− ζ1ζ2 ) (ζ1 − ζ2)2 + 4 . In both cases the desired separable potentials are equal to V (q1, q2) = F1(u1)− F2(u2) u1 − u2 , u1,2 = ζ1,2 or u1,2 = ζ̂1,2. Polar coordinates. Using one nontrivial eigenvalue of characteristic tensors K = K(2)g−1 and K̂ = K(2)ĝ−1 we can introduce only one coordinate r = √ q21 + q22, or r̂ = √ q21 + q22 1 + q21 + q22 . The second coordinate is a function on q1/q2, for instance, it could be an angle ϕ = arctan q1/q2, because solutions of the compatibility equations (2.11) and (2.15) have the following form V (q1, q2) = F1(ρ) + F2(ϕ) ρ2 , ρ = r or ρ = r̂. Cartesian coordinates. Separable in the Cartesian coordinates solution of the compatibility condition (2.11) reads as V (q1, q2) = F1(q1) + F2(q2). (2.22) In nonholonomic case characteristic tensor K̂ = K(1)ĝ−1 yields only one coordinate %̂ = √ 1 + q21 1 + q21 + q22 and the separable solution of the compatibility condition (2.15) looks like V (q1, q2) = F1(%̂) + F2(φ̂) %̂2 , φ̂ = arctan q1. Thus, after nonholonomic reduction one gets separable potential, which is different, even in the form of standard potential (2.22) separable in Cartesian coordinates. Summing up, we have found four characteristic coordinate systems for the Bertrand–Darboux type equation (2.18) associated with the integrable non-Hamiltonian vector field X̂ (2.7). Ad- ditional first integral of this field X̂ is a polynomial of the second order in momenta defined by the standard Killing tensor on the plane. Consequently, we can directly apply the standard Stäckel theory to the nonholonomic Heisenberg type systems. Integrability of Nonholonomic Heisenberg Type Systems 9 3 Stäckel systems In Hamiltonian case there is a one-to-one correspondence between the so-called Stäckel systems, integrable Killing tensors which mutually commute in the algebraic sense and separation of variables [3, 14]. The nondegenerate n× n Stäckel matrix S, which j column depends only on variable uj detS 6= 0, ∂Skj ∂um = 0, j 6= m defines n functionally independent integrals of motion Hk = n∑ j=1 Cjk ( p2uj + Uj(uj) ) , C = S−1, (3.1) which are in involution with respect to canonical Poisson brackets {ui, puj} = δij , {ui, uj} = {pui , puj} = 0. The common level surface of the first integrals H1 = α1, . . . ,H2 = αn is diffeomorphic to the n-dimensional real torus and one immediately gets p2uj = n∑ k=1 αkSkj(uj)− Uj(uj). It allows us to calculate quadratures for the corresponding Hamiltonian vector field X = PdH1: n∑ j=1 ∫ γj(puj ,uj) γ0(p0,q0) Skj(u)du√ n∑ k=1 αkSkj(u)− Uj(u) = βk, k = 1, . . . , n, (3.2) where β1 = t and β2, . . . , βn are constants of integration. Solution of the problem is thus reduced to solving a sequence of one-dimensional problems, which is the essence of the method of separation of variables [3]. 3.1 Stäckel matrices for reduced systems In the definition of the Stäckel integrals of motion (3.1) momenta pu have to be canonically conjugated to eigenvalues u of the characteristic (1, 1) tensor K. For instance, if we take ten- sor K(3) (2.19) the eigenvalues of K are parabolic coordinates ζ1,2 (2.20) and the corresponding momenta read as pζ1,2 = p1 2 ± √ q21 + q22 ∓ q1 2q2 p2. In nonholonomic case the eigenvalues of K̂ are given by ζ̂1,2 (2.21) and the conjugated momenta have the following form pζ̂1,2 = q22 + 1± q1 √ 1 + q22 √ q21 + q22 2 ( 1− q21 ) p1 − q1 ∓ √ 1 + q22 √ q21 + q22 2q2 p2. In similar manner we can calculate all the canonical variables associated with tensors K(j) (2.19) and to obtain standard Stäckel matrices in Hamiltonian case S(1) = ( 1 1 1 −1 ) , S(2) = ( 0 1 1 −r−2 ) , S(3) = ( 1 1 −ζ−11 −ζ−12 ) 10 Yu.A. Grigoryev, A.P. Sozonov and A.V. Tsiganov and S(4) =  1 ξ1 − κ2 1 ξ2 − κ2 −1 ξ1(ξ1 − κ2) −1 ξ2(ξ2 − κ2)  . For the nonholonomic Heisenberg type systems associated with tensors K(j) (2.19) Stäckel mat- rices are equal to Ŝ(1) = 0 1 %̂2(1− %̂2) 1 −1 %̂4(1− %̂2)  , Ŝ(2) = 0 1 1− r̂2 1 −1 r̂2(1− r̂2)  and Ŝ(3) =  1 1− ζ̂1 1 1− ζ̂2 −1 ζ̂1(1− ζ̂1) −1 ζ̂2(1− ζ̂2)  , Ŝ(4) =  1 (ξ̂1 − κ2)(1− ξ̂1) 1 (ξ̂2 − κ2)(1− ξ̂2) −1 ξ̂1(ξ̂1 − κ2)(1− ξ̂1) −1 ξ̂2(ξ̂2 − κ2)(1− ξ̂2)  . It is easy to see, that imposing linear, non integrable constraint (1.7) in polar, parabolic and ellip- tic cases we have to multiply j column of the standard Stäckel matrices on the function (1−uj), where uj is the eigenvalue of K̂. Similar transformations of the Stäckel matrices for nonholomic systems on a two-dimensional sphere are discussed in [22]. Using a new time variable τ defined by dt = µ−1dτ ≡ ( 1 + q21 + q22 ) dτ we can rewrite the conformally Hamiltonian vector field X̂ (2.7) in the Hamiltonian form and obtain quadratures similar to (3.2). Thus, in nonholonomic case solution of the problem is reduced to solving a sequence of one-dimensional problems after suitable change of time. 3.2 Compatible Poisson brackets Riemannian geometry is not, a priori, concerned with symplectic or Poisson structures. Never- theless, it is known that any tensor field L with vanishing Nijenhuis torsion on the Riemannian manifold Q yields trivial deformation of the canonical Poisson bracket {·, ·} on its cotangent bundle T ∗Q {qi, qj}L = 0, {qi, pj}L = −Lij , {pi, pj}L = ( ∂Lkj ∂qi − ∂Lki ∂qj ) pk, (3.3) where (pi, qi) are fibered coordinates. This Poisson bracket is compatible with the canonical one, so that there is a recursion operator N = PLP −1, which allows us to construct a whole family of compatible brackets associated with Poisson bivectors P (m) L = NmP on T ∗Q [21, 22]. Integrability of Nonholonomic Heisenberg Type Systems 11 According to [3] generic solution of the Killing equation K (2.16) determines a conformal Killing tensor with vanishing Nijenhuis torsion L = K − tr(K)g and Turiel’s deformation (3.3) of the canonical Poisson bracket [21, 22]. The eigenvalues of the corresponding recursion operator N = PLP −1 are characteristic coordinates. Integrals of motion H1 (2.8) and H2 (2.9) are in involution with respect to the compatible Poisson brackets {H1, H2} = {H1, H2}L = 0 and to other polynomial Poisson brackets associated with the Poisson bivectors P (m) L . Indeed, we can find such L-tensors directly from the Hamilton function H1 using modern software [16]. For the Heisenberg type systems, we have to solve the same Killing equation (2.14) and second equation on potential with (1, 1) tensor K̂ = Kĝ. This tensor does not appear in the Turiel construction of the Poisson bracket and we can directly prove the following proposition. Proposition 3. Integrals of motion Ĥ1 (2.12) and Ĥ2 (2.13) are in involution{ Ĥ1, Ĥ2 }∧ = { Ĥ1, Ĥ2 } L = 0 with respect to the compatible Poisson brackets {·, ·}∧ (2.5) and {·, ·}L (3.3). The eigenvalues of the recursion operator N̂ = PLP̂ −1, where P̂ is given by (2.7), are characteristic coordinates discussed in the previous Section. The proof consists of the straightforward calculations. Thus, starting with a common Poisson bivector PL we can construct two recursion opera- tors N and N̂ and two families of the Poisson brackets on T ∗R2. 3.3 Lax matrices We can construct Lax representations for the Stäckel systems with uniform rational and poly- nomial potentials Uj = U in (3.1), see [13] and references within. For the sake of brevity we consider only systems associated with tensor K(3) (2.19). If V (q1, q2) = 0, we can introduce a well-known 2× 2 Lax matrix for the geodesic motion on the plane L(u) = ( h e f −h ) (u) ≡  1 2 de(u) dt e(u) −1 2 d2e(u) dt2 −1 2 de(u) dt  , (3.4) which satisfies to the Lax equation d dt L = [L,A] ≡ LA−AL, with A = ( 0 1 0 0 ) . (3.5) This equation guarantees that the eigenvalues of L are conserved quantities in the involution. In our case function e(u) depends on the so-called spectral parameter u e(u) = (u− ζ1)(u− ζ2) u = u− 2q1 − q22 u , 12 Yu.A. Grigoryev, A.P. Sozonov and A.V. Tsiganov which is completely defined by the eigenvalues of the characteristic tensor K = K(3)g−1 or parabolic coordinates ζ1,2 on the plane. The involution property of eigenvalues of the Lax matrix L is equivalent to existence of a classical r-matrix r12, that as {L1(u),L2(v)} = [r12(u, v),L1(u)]− [r21(u, v),L2(v)]. Here we use the familiar notation for tensor product of L and unit matrix I L1(u) = L(u)⊗ I, L2(v) = I⊗ L(v), r21(u, v) = Πr12(v, u)Π, whereas Π is the permutation operator: Πx ⊗ y = y ⊗ x, ∀x, y. Evaluating canonical Poisson brackets (2.3) between entries of the Lax matrix L(u) (3.4) we obtain a constant r-matrix r12(u, v) = 2 v − u Π ≡ 2 v − u  1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1  . (3.6) If we want to consider separable in parabolic coordinates potentials V (q1, q2), we have to add some items to the initial Lax matrix L(u) (3.4). For instance, we can take the following additive perturbation LV (u) = L(u) + ( 0 0 ∆f 0 ) , ∆f = [ φ(u)e−1(u) ] , where φ(u) is an arbitrary constant function and [g(u)] is a truncated Laurent series expansion of g(u) with respect to the variable u. For instance, when φ(u) = −au3 the Taylor part of expansion about u =∞ looks like[ −au3e−1(u) ] = −a ( u2 + 2uq1 + q22 + 4q21 ) . Substituting this expression into LV (u) one gets a Lax matrix for the Hénon–Heiles system on the plane with the Hamiltonian H1 = 1 2 ( p21 + p22 ) + 8aq1 ( 2q21 + q22 ) . In nonholonomic case we can also find similar Lax matrices, classical r-matrix and integrable potentials. Proposition 4. For the Heisenberg type systems equations of motion (2.7) when V (q1, q2) = 0 can be rewritten in the Lax form (3.5) if L̂(u) = ( ĥ ê f̂ −ĥ ) (u) ≡  1 2µ dê(u) dt ê(u) − d µdt ( 1 2µ dê(u) dt ) − 2Ĥ1ê(u) − 1 2µ dê(u) dt  (3.7) and Â = ( 0 µ −2µĤ1 0 ) . Here µ = ( 1 + q21 + q22 )−1 is a conformal factor (2.7) and function ê(u) = ( u− ζ̂1 )( u− ζ̂2 ) u = u− 2q1 1 + q21 + q22 − q22 u ( 1 + q21 + q22 ) is completely defined by characteristic coordinates ζ̂1,2 (2.21). Integrability of Nonholonomic Heisenberg Type Systems 13 The Lax matrix (3.7) was constructed by using the generic construction of the Lax matrices for the Stäckel systems [13]. The modification consists only of application of the nontrivial conformal factor µ. Evaluating Poisson brackets {·, ·}∧ (2.5) between entries of the Lax matrix L̂(u) (3.7) we find the corresponding classical r-matrix r̂12(u, v) = r12(u, v) + 2  0 0 0 0 0 0 ê(v) 0 0 0 0 0 −f̂(v) 0 0 0  , (3.8) which explicitly depends on dynamical variables via entries of L̂(u) (3.7). As above, we can consider additive perturbation of this Lax matrix L̂V (u) = L̂(u) + ( 0 0 ∆f̂ 0 ) , ∆f̂ = ( u2 − 1 )[ φ(u)ê−1(u) ] in order to get Lax matrices for the nonholonomic Heisenberg type systems with V 6= 0. For instance, when φ(u) = −au3 the Teylor part of expansion about u =∞ looks like [ −au3ê−1(u) ] = −a ( u2 + 2uq1 1 + q21 + q22 + q22 1 + q21 + q22 + 4q21( 1 + q21 + q22 )2 ) . Substituting this expression into L̂V (u) one gets a Lax matrix for the nonholonomic counterpart of the Hénon–Hieles system on the plane with the Jacobi integral Ĥ1 = 1 2 ( p21 + p22 + (q2p1 − q1p2)2 ) + 8aq1 ( q21q 2 2 + q42 + 2q21 + q22 )( 1 + q21 + q22 )3 . 4 Conclusion Imposing nonholonomic constraints to Hamiltonian systems and making a suitable reduction one gets some special class of non Hamiltonian systems on the reduced phase space, see book [7]. Usually we can not investigate the reduced system using standard mathematical methods of Hamiltonian dynamics. For instance, we do not know how to get Lax matrices, classical r- matrices or compatible Poisson brackets for the conformally Hamiltonian systems associated with the nonholonomic Chaplygin ball, nonholonomic Suslov or Veselova systems. In this note we find Killing tensors and compatible Poisson brackets, describe integrable potentials and characteristic coordinates, evaluate Stäckel matrices and Stäckel quadratures, to show Lax matrices and classical r-matrices for the nonholonomic Heisenberg type systems. Indeed, we prove that some modern geometric methods of Hamiltonian mechanics can be directly applied to the reduced conformally Hamiltonian systems. Other similar examples can be found in [6, 8, 20, 22, 23]. Of course, we can impose other nonholonomic constraints on the original Hamiltonian sys- tem (1.1), (1.2). It may be interesting to describe all the constraints which lead to nontrivial integrable metrics and potentials on the reduced phase space. Acknowledgements We are very grateful to the referees for thorough analysis of the manuscript, constructive sug- gestions and proposed corrections, which certainly lead to a more profound discussion of the results. We are also deeply grateful A.V. Borisov and I.A. Bizayev for the relevant discussion. 14 Yu.A. Grigoryev, A.P. Sozonov and A.V. Tsiganov Section 2 was written by A.V. Tsiganov and supported by the Russian Science Foundation (project 15-12-20035). Section 3 was written by Yu.A. Grigoryev and A.P. Sozonov within the framework of the Russian Science Foundation (project 15-11-30007). References [1] Ankiewicz A., Pask C., The complete Whittaker theorem for two-dimensional integrable systems and its application, J. Phys. A: Math. Gen. 16 (1983), 4203–4208. [2] Bates L., Śniatycki J., Nonholonomic reduction, Rep. Math. Phys. 32 (1993), 99–115. [3] Benenti S., Orthogonal separable dynamical systems, in Differential Geometry and its Applications (Opava, 1992), Math. Publ., Vol. 1, Editors O. Kowalsky, D. Krupka, Silesian University Opava, Opava, 1993, 163– 184. [4] Bertrand J.M., Mémoire sur quelques-unes des forms les plus simples que puissent présenter les intégrales des équations différentielles du mouvement d’un point matériel, J. Math. Pures Appl. 2 (1857), 113–140. [5] Bizyaev I.A., Borisov A.V., Mamaev I.S., Hamiltonization of elementary nonholonomic systems, Russ. J. Math. Phys. 22 (2015), 444–453, arXiv:1601.00884. [6] Bizyaev I.A., Tsiganov A.V., On the Routh sphere problem, J. Phys. A: Math. Theor. 46 (2013), 085202, 11 pages, arXiv:1210.7903. [7] Bloch A.M., Nonholonomic mechanics and control, Interdisciplinary Applied Mathematics, Vol. 24, Springer- Verlag, New York, 2003. [8] Borisov A.V., Mamaev I.S., Symmetries and reduction in nonholonomic mechanics, Regul. Chaotic Dyn. 20 (2015), 553–604. [9] Borisov A.V., Mamaev I.S., Bizyaev I.A., The hierarchy of dynamics of a rigid body rolling without slipping and spinning on a plane and a sphere, Regul. Chaotic Dyn. 18 (2013), 277–328. [10] Borisov A.V., Mamaev I.S., Tsiganov A.V., Non-holonomic dynamics and Poisson geometry, Russ. Math. Surv. 69 (2014), 481–538. [11] Cushman R., Duistermaat H., Śniatycki J., Geometry of nonholonomically constrained systems, Advanced Series in Nonlinear Dynamics, Vol. 26, World Sci. Publ. Co. Pte. Ltd., Hackensack, NJ, 2010. [12] Darboux G., Sur un probléme de mécanique, Arch. Néerl. 6 (1901), 371–376. [13] Eilbeck J.C., Enol’skii V.Z., Kuznetsov V.B., Tsiganov A.V., Linear r-matrix algebra for classical separable systems, J. Phys. A: Math. Gen. 27 (1994), 567–578, hep-th/9306155. [14] Eisenhart L.P., Separable systems of Stäckel, Ann. of Math. 35 (1934), 284–305. [15] Fassò F., Sansonetto N., Conservation of energy and momenta in nonholonomic systems with affine con- straints, Regul. Chaotic Dyn. 20 (2015), 449–462, arXiv:1505.01172. [16] Grigoryev Yu.A., Tsiganov A.V., Symbolic software for separation of variables in the Hamilton–Jacobi equation for the L-systems, Regul. Chaotic Dyn. 10 (2005), 413–422, nlin.SI/0505047. [17] Kozlov V.V., The Euler–Jacobi–Lie integrability theorem, Regul. Chaotic Dyn. 18 (2013), 329–343. [18] Molina-Becerra M., Galán-Vioque J., Freire E., Dynamics and bifurcations of a nonholonomic Heisenberg system, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 22 (2012), 1250040, 14 pages. [19] Smirnov R.G., On the classical Bertrand–Darboux problem, J. Math. Sci. 151 (2008), 3230–3244, math-ph/0604038. [20] Tsiganov A., Integrable Euler top and nonholonomic Chaplygin ball, J. Geom. Mech. 3 (2011), 337–362, arXiv:1002.1123. [21] Tsiganov A.V., On bi-integrable natural Hamiltonian systems on Riemannian manifolds, J. Nonlinear Math. Phys. 18 (2011), 245–268, arXiv:1006.3914. [22] Tsiganov A.V., One family of conformally Hamiltonian systems, Theoret. Math. Phys. 173 (2012), 1481– 1497, arXiv:1206.5061. [23] Tsiganov A.V., On integrable perturbations of some nonholonomic systems, SIGMA 11 (2015), 085, 19 pages, arXiv:1505.01588. [24] van der Schaft A.J., Maschke B.M., On the Hamiltonian formulation of nonholonomic mechanical systems, Rep. Math. Phys. 34 (1994), 225–233. [25] Whittaker E.T., A treatise on the analytical dynamics of particles and rigid bodies, Cambridge Mathematical Library , Cambridge University Press, Cambridge, 1988. http://dx.doi.org/10.1088/0305-4470/16/18/021 http://dx.doi.org/10.1016/0034-4877(93)90073-N http://dx.doi.org/10.1134/S1061920815040032 http://dx.doi.org/10.1134/S1061920815040032 http://arxiv.org/abs/1601.00884 http://dx.doi.org/10.1088/1751-8113/46/8/085202 http://arxiv.org/abs/1210.7903 http://dx.doi.org/10.1007/b97376 http://dx.doi.org/10.1134/S1560354715050044 http://dx.doi.org/10.1134/S1560354713030064 http://dx.doi.org/10.1070/RM2014v069n03ABEH004899 http://dx.doi.org/10.1070/RM2014v069n03ABEH004899 http://dx.doi.org/10.1088/0305-4470/27/2/038 http://arxiv.org/abs/hep-th/9306155 http://dx.doi.org/10.2307/1968433 http://dx.doi.org/10.1134/S1560354715040048 http://arxiv.org/abs/1505.01172 http://dx.doi.org/10.1070/RD2005v010n04ABEH000323 http://arxiv.org/abs/nlin.SI/0505047 http://dx.doi.org/10.1134/S1560354713040011 http://dx.doi.org/10.1142/S021812741250040X http://dx.doi.org/10.1007/s10958-008-9036-0 http://arxiv.org/abs/math-ph/0604038 http://dx.doi.org/10.3934/jgm.2011.3.337 http://arxiv.org/abs/1002.1123 http://dx.doi.org/10.1142/S1402925111001507 http://dx.doi.org/10.1142/S1402925111001507 http://arxiv.org/abs/1006.3914 http://dx.doi.org/10.1007/s11232-012-0128-0 http://arxiv.org/abs/1206.5061 http://dx.doi.org/10.3842/SIGMA.2015.085 http://arxiv.org/abs/1505.01588 http://dx.doi.org/10.1016/0034-4877(94)90038-8 http://dx.doi.org/10.1017/CBO9780511608797 http://dx.doi.org/10.1017/CBO9780511608797 1 Introduction 1.1 Main definitions 2 Reduced systems on the plane 2.1 Bertrand–Darboux type equation 2.2 Solutions of the Bertrand–Darboux type equation 3 Stäckel systems 3.1 Stäckel matrices for reduced systems 3.2 Compatible Poisson brackets 3.3 Lax matrices 4 Conclusion References

Integrability of Nonholonomic Heisenberg Type Systems

Institution

Ähnliche Einträge