7 research outputs found
Conservation of `moving' energy in nonholonomic systems with affine constraints and integrability of spheres on rotating surfaces
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
Conservation of energy and momenta in nonholonomic systems with affine constraints
We characterize the conditions for the conservation of the energy and of the
components of the momentum maps of lifted actions, and of their `gauge-like'
generalizations, in time-independent nonholonomic mechanical systems with
affine constraints. These conditions involve geometrical and mechanical
properties of the system, and are codified in the so-called
reaction-annihilator distribution
A Geometric Characterization of Certain First Integrals for Nonholonomic Systems with Symmetries
We study the existence of first integrals in nonholonomic systems with symmetry. First we define the concept of M -cotangent lift of a vector field on a manifold Q in order to unify the works [Balseiro P., Arch. Ration. Mech. Anal. 214 (2014), 453-501, arXiv:1301.1091], [Fosse F., Ramos A., Sansonetto N., Regul. Chaotic Dyn. 12 (2007), 579588], and [Fosse F., Giacobbe A., Sansonetto N., Rep. Math. Phys. 62 (2008), 345-367]. Second, we study gauge symmetries and gauge momenta, in the cases in which there are the symmetries that satisfy the so-called vertical symmetry condition. Under such condition we can predict the number of linearly independent first integrals (that are gauge momenta). We illustrate the theory with two examples
Moving energies as first integrals of nonholonomic systems with affine constraints
In nonholonomic mechanical systems with constraints that are affine (linear
nonhomogeneous) functions of the velocities, the energy is typically not a
first integral. It was shown in [Fass\`o and Sansonetto, JNLS, 26, (2016)]
that, nevertheless, there exist modifications of the energy, called there
moving energies, which under suitable conditions are first integrals. The first
goal of this paper is to study the properties of these functions and the
conditions that lead to their conservation. In particular, we enlarge the class
of moving energies considered in [Fass\`o and Sansonetto, JNLS, 26, (2016)].
The second goal of the paper is to demonstrate the relevance of moving energies
in nonholonomic mechanics. We show that certain first integrals of some well
known systems (the affine Veselova and LR systems), which had been detected on
a case-by-case way, are instances of moving energies. Moreover, we determine
conserved moving energies for a class of affine systems on Lie groups that
include the LR systems, for a heavy convex rigid body that rolls without
slipping on a uniformly rotating plane, and for an -dimensional
generalization of the Chaplygin sphere problem to a uniformly rotating
hyperplane.Comment: 25 pages, 1 figure. Final version prepared according to the
modifications suggested by the referees of Nonlinearit