Geometry of Invariant Tori of Certain Integrable Systems with Symmetry and an Application to a Nonholonomic System

Fassò, Francesco; Giacobbe, Andrea

doi:10.3842/sigma.2007.051

Cited by 8 publications

(25 citation statements)

References 15 publications

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“…Since the symmetry group is compact and acts freely on M * 0 , this in turn implies that the unreduced dynamics in M * 0 is quasi-periodic on tori of dimension up to three. This was proven in [16] using a reconstruction result from periodic dynamics, originally due to Field and Krupa (see particularly [18,14,16,9,8,15]). Reference [27] reaches the same conclusion, but restricted to the motion of the center of mass, that undergoes quasi-periodic motions on tori of dimension up to two.…”

Section: Integrability Of a Sphere Rolling On A Rotating Surface Of Rmentioning

confidence: 92%

“…The proof given above shows that each relative periodic orbit is fibered by tori of some dimension between 1 and 3, but it does not ensure that this dimension is the same across different relative periodic orbits and that the invariant tori are the fibers of a fibration of (an open subset of) the phase space. This property is important, because it implies the existence of the appropriate number of first integrals that are usually associated to integrability (for some results on this point in the case Ω = 0 see [9]).…”

Section: Integrability Of a Sphere Rolling On A Rotating Surface Of Rmentioning

confidence: 99%

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Conservation of ‘Moving’ Energy in Nonholonomic Systems with Affine Constraints and Integrability of Spheres on Rotating Surfaces

Fassò

Sansonetto

2016

J Nonlinear Sci

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Energy is in general not conserved for mechanical nonholonomic systems with affine constraints. In this article we point out that, nevertheless, in certain cases, there is a modification of the energy that is conserved. Such a function coincides with the energy of the system relative to a different reference frame, in which the constraint is linear. After giving sufficient conditions for this to happen, we point out the role of symmetry in this mechanism. Lastly, we apply these ideas to prove that the motions of a heavy homogeneous solid sphere that rolls inside a convex surface of revolution in uniform rotation about its vertical figure axis, are (at least for certain parameter values and in open regions of the phase space) quasi-periodic on tori of dimension up to three.MSC: 70F25, 37J60, 37J15, 70E18 * This work is part of the research projects Symmetries and integrability of nonholonomic mechanical systems of the University of Padova and PRIN Teorie geometriche e analitiche dei sistemi Hamiltoniani in dimensioni finite e infinite.

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Section: Integrability Of a Sphere Rolling On A Rotating Surface Of Rmentioning

confidence: 92%

Section: Integrability Of a Sphere Rolling On A Rotating Surface Of Rmentioning

confidence: 99%

Conservation of ‘Moving’ Energy in Nonholonomic Systems with Affine Constraints and Integrability of Spheres on Rotating Surfaces

Fassò

Sansonetto

2016

J Nonlinear Sci

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“…Details on these results, and their modifications under more general hypotheses, can be found in the quoted references and in the recent books [14,8]. The related problem of reconstructing families of relative periodic orbits (e.g., when the entire reduced dynamics is periodic) has been studied in [15] in the case of a free action of a compact group (see also [9,8]).…”

Section: Relative Quasi-periodic Torimentioning

confidence: 99%

Quasi-periodicity in relative quasi-periodic tori

2015

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At variance from the cases of relative equilibria and relative periodic orbits of dynamical systems with symmetry, the dynamics in relative quasi-periodic tori (namely, subsets of the phase space that project to an invariant torus of the reduced system on which the flow is quasi-periodic) is not yet completely understood. Even in the simplest situation of a free action of a compact and abelian connected group, the dynamics in a relative quasi-periodic torus is not necessarily quasi-periodic. It is known that quasi-periodicity of the unreduced dynamics is related to the reducibility of the reconstruction equation, and sufficient conditions for it are virtually known only in a perturbation context. We provide a different, though equivalent, approach to this subject, based on the hypothesis of the existence of commuting, group-invariant lifts of a set of generators of the reduced torus. Under this hypothesis, which is shown to be equivalent to the reducibility of the reconstruction equation, we give a complete description of the structure of the relative quasi-periodic torus, which is a principal torus bundle whose fibers are tori of a dimension which exceeds that of the reduced torus by at most the rank of the group. The construction can always be done in such a way that these tori have minimal dimension and carry ergodic flow.1 Sometimes a different convention is used, and the terms relative equilibria and relative periodic orbits denote the individual orbits in such relative sets, rather than the relative sets themselves.

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“…Since the sheaf of 2-forms on P vanishing on F is fine (cf. [31]), for each j, let ζ j be a 2-form defined on U j and vanishing on F | Uj such that (19) ζ l − ζ j = dκ jl (cf. proof of Proposition 5).…”

Section: Corollarymentioning

confidence: 99%

“…Thus far there are two different approaches to the study of the integrability outside the (symplectic or Poisson) Hamiltonian framework: 1) reconstruction from reduced periodic (and recently quasi-periodic) dynamics (cf. [1,5,18,19,21,33]); 2) possible generalisations of the Liouville-Arnol'd Theorem or its non-commutative counterpart both in symplectic and Poisson geometry (cf. [6,3,11,17,22]).…”

Section: Introductionmentioning

confidence: 99%

Twisted isotropic realisations of twisted Poisson structures

Sansonetto¹,

Sepe²

2013

Journal of Geometric Mechanics

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Motivated by the recent connection between nonholonomic integrable systems and twisted Poisson manifolds made in [3], this paper investigates the global theory of integrable Hamiltonian systems on almost symplectic manifolds as an initial step to understand Hamiltonian integrability on twisted Poisson (and Dirac) manifolds. Non-commutative integrable Hamiltonian systems on almost symplectic manifolds were first defined in [17], which proved existence of local generalised action-angle coordinates in the spirit of the Liouville-Arnol'd theorem. In analogy with their symplectic counterpart, these systems can be described globally by twisted isotropic realisations of twisted Poisson manifolds, a special case of symplectic realisations of twisted Dirac structures considered in [8]. This paper classifies twisted isotropic realisations up to smooth isomorphism and provides a cohomological obstruction to the construction of these objects, generalising some of the main results of [13]

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Geometry of Invariant Tori of Certain Integrable Systems with Symmetry and an Application to a Nonholonomic System

Cited by 8 publications

References 15 publications

Conservation of ‘Moving’ Energy in Nonholonomic Systems with Affine Constraints and Integrability of Spheres on Rotating Surfaces

Conservation of ‘Moving’ Energy in Nonholonomic Systems with Affine Constraints and Integrability of Spheres on Rotating Surfaces

Quasi-periodicity in relative quasi-periodic tori

Twisted isotropic realisations of twisted Poisson structures

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