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 is the pull-back of the energy of the system written in a system of time-dependent coordinates in which the constraint is linear, and for this reason will be called a 'moving' energy. After giving sufficient conditions for the existence of a conserved, time-independent moving energy, 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.

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

Sansonetto, Nicola
2016-01-01

Abstract

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 is the pull-back of the energy of the system written in a system of time-dependent coordinates in which the constraint is linear, and for this reason will be called a 'moving' energy. After giving sufficient conditions for the existence of a conserved, time-independent moving energy, 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.
Nonholonomic mechanical systems; Conservation of energy; Rolling rigid bodies; Symmetries and momentum maps; Integrability
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11562/1010108
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