Nonholonomic systems via moving frames: Cartan equivalence and Chaplygin Hamiltonization

Abstract

A nonholonomic system, for short “NH,” consists of a configuration space Q ⁿ, a Lagrangian $L(q,\dot q,t)$, a nonintegrable constraint distribution $\mathcal{H} \subset TQ$, with dynamics governed by Lagrange-d’Alembert’s principle. We present here two studies, both using adapted moving frames. In the first we explore the affine connection viewpoint. For natural Lagrangians L = T − V, where we take V = 0 for simplicity, NH-trajectories are geodesics of a (nonmetric) connection ∇_{N H} which mimics Levi-Civita’s. Local geometric invariants are obtained by Cartan’s method of equivalence. As an example, we analyze Engel’s (2–4) distribution. This is the first such study for a distribution that is not strongly nonholonomic. In the second part we study G-Chaplygin systems; for those, the constraints are given by a connection φ: T Q → Lie(G) on a principal bundle G ↪ Q → S = Q/G and the Lagrangian L is G-equivariant. These systems compress to an almost Hamiltonian system (T*S, H ^φ, Ω_{N H}), Ω^{N H} = Ω_can + (J.K), with d(J.K) ≠ = 0 in general; the momentum map J : T*Q → Lie(G) and the curvature form K : T Q → Lie(G)* are matched via the Legendre transform. Under an s ε S dependent time reparametrization, a number of compressed systems become Hamiltonian, i.e., Ω_{N H} is sometimes conformally symplectic. Anecessary condition is the existence of an invariant volume for the original system. Its density produces a candidate for conformal factor. Assuming an invariant volume, we describe the obstruction to Hamiltonization. An example of a Hamiltonizable system is the “rubber” Chaplygin’s sphere, which extends Veselova’s system in T*S O(3). This is a ball with unequal inertia coefficients rolling without slipping on the plane, with vertical rotations forbidden. Finally, we discuss reduction of internal symmetries. Chaplygin’s “marble,” where vertical rotations are allowed, is not Hamiltonizable at the compressed T*S O(3) level. We conjecture that it is also not Hamiltonizable when reduced to T*S ².

The authors thank the Brazilian funding agencies CNPq and FAPERJ: a CNPq research fellowship (JK), a CNPq post-doctoral fellowship at Berkeley (PMR), a FAPERJ visiting fellowship to Rio de Janeiro (KE). (JK) thanks the E. Schrödinger Institute, Vienna, for financial support during Alanfest and the Poisson Geometry Program, August 2003.