The Bouncing Penny and Nonholonomic Impacts

Abstract

The evolution of a Lagrangian mechanical system is variational. Likewise, when dealing with a hybrid Lagrangian system (a system with discontinuous impacts), the impacts can also be described by variations. These variational impacts are given by the so-called Weierstrass-Erdmann corner conditions. Therefore, hybrid Lagrangian systems can be completely understood by variational principles.Unlike typical (unconstrained / holonomic) Lagrangian systems, nonholonomically constrained Lagrangian systems are not variational. However, by using the Lagrange-d'Alembert principle, nonholonomic systems can be described as projections of variational systems. This paper works out the analogous version of the Weierstrass-Erdmann corner conditions for nonholonomic systems and examines the billiard problem with a rolling disk.

Invariant Forms in Hybrid and Impact Systems and a Taming of Zeno

“…Condition (10) requires Cartan's magic formula: (11) holds because pullbacks distribute over the wedge product: (10), we see that…”

“…However, we can instead utilize the Lagrange-d'Alembert principle. This leads to a modified version of ( 4), [11]:…”

“…If we denote A := {α ∈ Ω(M ) : L X α = 0}, then it is already known that A ⊂ Ω(M ) is a ∧-subalgebra closed under d and i X (see Corollary 3.4.5 in [1]). Therefore, in order to prove the theorem, it suffices only to check (10) and (11). Let α, β ∈ A H .…”

“…Consider i X α. This satisfies (11) because α satisfies (10) and ( 10) is satisfied because i X i X α = 0.…”

Symplectic and Cosymplectic Reduction for simple hybrid forced mechanical systems with symmetries

“…This type of hybrid system has been mainly employed for the understanding of locomotion gaits in bipeds and insects [3], [27], [42]. In the situation where the vector field X is associated with a mechanical system (Lagrangian or Hamiltonian), alternative approaches for mechanical systems with nonholonomic and unilateral constraints have been considered in [8], [13], [14], [28], [29].…”