Nonholonomic systems via moving frames: Cartan equivalence and Chaplygin Hamiltonization

Abstract

A nonholonomic system consists of a configuration space Q, a Lagrangian L, and an nonintegrable constraint distribution H, with dynamics governed by Lagrange-d'Alembert's principle. We present two studies both using adapted moving frames. In the first study we apply Cartan's method of equivalence to investigate the geometry underlying a nonholonomic system. As an example we compute the differential invariants for a nonholonomic system on a four-dimensional configuration manifold endowed with a rank two (Engel) distribution. In the second part we study G-Chaplygin systems. These are systems where the constraint distribution is given by a connection on a principal fiber bundle with total space Q and base space S=Q/G, and with a G-equivariant Lagrangian. These systems compress to an almost Hamiltonian system on $T^{\ast }S$. Under an $s\in S$ dependent time reparameterization a number of compressed systems become Hamiltonian. A necessary condition for Hamiltonization is the existence of an invariant measure on the original system. Assuming an invariant measure we describe the obstruction to Hamiltonization. Chaplygin's "rubber" sphere, a ball with unequal inertia coefficients rolling without slipping or spinning (about the vertical axis) on a plane is Hamiltonizable when compressed to $T^{\ast }SO(3)$. Finally we discuss reduction of internal symmetries. Chaplygin's "marble" (where spinning is allowed) is not Hamiltonizable when compressed to $T^{\ast }SO(3)$; we conjecture that it is also not Hamiltonizable when reduced to $T^{\ast }{S}^2$.

Dedicated to Alan Weinstein on his 60th birthday, 39 pages

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• Gauge momenta as Casimir functions of nonholonomic systems
• Hamiltonisation, measure preservation and first integrals of the multi-dimensional rubber Routh sphere
• Hamiltonization of solids of revolution through reduction
• Extension Phenomena for Holomorphic Geometric Structures
• Generalisation of Chaplygin's Reducing Multiplier Theorem with an application to multi-dimensional nonholonomic dynamics
• Stochastic Chaplygin systems
• Reduction of a Hamilton-Jacobi equation for nonholonomic systems
• Geodesic Reduction via Frame Bundle Geometry