Unimodularity and preservation of volumes in nonholonomic mechanics

Abstract

The equations of motion of a mechanical system subjected to nonholonomic linear constraints can be formulated in terms of a linear almost Poisson structure in a vector bundle. We study the existence of invariant measures for the system in terms of the unimodularity of this structure. In the presence of symmetries, our approach allows us to give necessary and sufficient conditions for the existence of an invariant volume, that unify and improve results existing in the literature. We present an algorithm to study the existence of a smooth invariant volume for nonholonomic mechanical systems with symmetry and we apply it to several concrete mechanical examples.

37 pages, 3 figures; v3 includes several changes to v2 that were done in accordance to the referee suggestions

References in corpus (7)

• Nonholonomic LR systems as Generalized Chaplygin systems with an Invariant Measure and Geodesic Flows on Homogeneous Spaces
• The Rolling Motion of a Ball on a Surface. New Integrals and Hierarchy of Dynamics
• Rubber rolling over a sphere
• Linear almost Poisson structures and Hamilton-Jacobi equation. Applications to nonholonomic Mechanics
• The inhomogeneous Suslov problem
• Non-existence of an invariant measure for a homogeneous ellipsoid rolling on the plane
• Hamiltonian Dynamics on Lie Algebroids, Unimodularity and Preservation of Volumes

Cited by in corpus (11)

• Invariant measures for contact Hamiltonian systems: symplectic sandwiches with contact bread
• The inhomogeneous Suslov problem
• Rolling balls over spheres in R^n
• The Hojman Construction and Hamiltonization of Nonholonomic Systems
• Gyroscopic Chaplygin systems and integrable magnetic flows on spheres
• Generalisation of Chaplygin's Reducing Multiplier Theorem with an application to multi-dimensional nonholonomic dynamics
• Demchenko's nonholonomic case of gyroscopic ball rolling without sliding over a sphere after his 1923 Belgrade doctoral thesis
• Nonholonomic connections, time reparametrizations, and integrability of the rolling ball over a sphere
• Spherical and planar ball bearings -- nonholonomic systems with invariant measures
• Unimodularity and invariant volume forms for Hamiltonian dynamics on Poisson-Lie groups
• Spherical and Planar Ball Bearings -- a Study of Integrable Cases