Generalisation of Chaplygin’s reducing multiplier theorem with an application to multi-dimensional nonholonomic dynamics

Abstract

A generalisation of Chaplygin's Reducing Multiplier Theorem is given by providing sufficient conditions for the Hamiltonisation of Chaplygin nonholonomic systems with an arbitrary number r of degrees of freedom via Chaplygin's multiplier method. The crucial point in the construction is to add an hypothesis of geometric nature that controls the interplay between the kinetic energy metric and the non-integrability of the constraint distribution. Such hypothesis can be systematically examined in concrete examples, and is automatically satisfied in the case r = 2 encountered in the original formulation of Chaplygin's theorem. Our results are applied to prove the Hamiltonisation of a multi-dimensional generalisation of the problem of a symmetric rigid body with a flat face that rolls without slipping or spinning over a sphere.

The geometry of nonholonomic Chaplygin systems revisited

“…Moreover, in this reference it is also shown that if (H) holds, and the basic measure µ = exp(σ )ν is preserved by the reduced flow, then the system is Hamiltonisable with the time reparametrisation dt = exp(σ (1 − r))dτ. Proposition 3.26 below shows that these two hypothesis taken together are equivalent to the condition that the system is φ -simple with φ = σ (r − 1), so the Hamiltonisation result of [24] is a particular consequence of Corollary 3.22.…”

“…We mention that there is a close relation between T and the geometric formulation of nonholonomic systems in terms of linear almost Poisson brackets on vector bundles [26,40] (see also [21]). Although the tensor T appears in the previous works of Koiller [36] and Cantrijn et al [11] (with an alternative definition than the one that we present here), its dynamical relevance had not been fully appreciated until the recent work García-Naranjo [24] where sufficient conditions for Hamiltonisation were given in terms of the coordinate representation of T . This work continues the research started in [24] by providing a coordinate-free definition of the gyroscopic tensor (Definition 3.3), and studying in depth its role in the almost symplectic structure of the equations of motion, the conditions for the existence of an invariant measure, and the Hamiltonisation of Chaplygin systems.…”

Hamiltonisation, measure preservation and first integrals of the multi-dimensional rubber Routh sphere

“…We now present the reduced equations of motion of a Chaplygin system. Our exposition mainly follows García-Naranjo [21]. The distribution D interpreted as a principal connection on the principal bundle π ∶ Q → S, induces a Riemannian metric on S that will be denoted by ⟪⋅,⋅⟫ S .…”

“…The advantage of the formulation in [21] and [22] with respect to these references is that condition (1.1) can be systematically examined in concrete examples. The statement that a φ -simple Chaplygin system allows a Chaplygin Hamiltonisation is independent of the number of degrees of freedom of the problem, and may be interpreted as a generalisation of the celebrated Chaplygin's Reducing Multiplier Theorem [9] -whose applicability is restricted to systems whose shape space has dimension 2 -see the discussion in [21] and [22].…”

Structure preserving discretization of time-reparametrized Hamiltonian systems with application to nonholonomic mechanics

“…Our main motivation to consider this kind of equations comes from a remarkable class of nonholonomic mechanical systems with symmetry, commonly known as Hamiltonizable G-Chaplygin systems, whose reduced dynamics have this structure (see e.g. [10,38,13,12,5,16,21], and references therein).…”