First Integrals and symmetries of nonholonomic systems

Abstract

In nonholonomic mechanics, the presence of constraints in the velocities breaks the well-understood link between symmetries and first integrals of holonomic systems, expressed in Noether's Theorem. However there is a known special class of first integrals of nonholonomic systems generated by vector fields tangent to the group orbits, called horizontal gauge momenta, that suggest that some version of this link should still hold. In this paper we prove that, under certain conditions on the symmetry Lie group, the (nonholonomic) momentum map is conserved along the nonholonomic dynamics, thus extending Noether Theorem to the nonholonomic framework. Our analysis leads to a constructive method, with fundamental consequences to the integrability of some nonholonomic systems as well as their hamiltonization. We apply our results to three paradigmatic examples: the snakeboard, a solid of revolution rolling without sliding on a plane and a heavy homogeneous ball that rolls without sliding inside a convex surface of revolution. In particular, for the snakeboard we show the existence of a new horizontal gauge momentum that reveals new aspects of its integrability. Contents

Linear connections and shape maps for second order ODEs with and without constraints

“…Example 5.2 (Ball rolling on a spherical surface). Consider a ball of radius r and mass m that is rolling without sliding on the inner side of half a sphere of radius R + r. We can take coordinates (x, y) for the centre of the ball, which moves on a half sphere Σ of radius R, with z = − R 2 − x 2 − y 2 + R and we can take Euler angles for the orientation of the ball, that is, as local coordinates for SO(3) (see [3,Section 5.3] and [23]). The configuration space is SO(3) × Σ.…”