Moving frames for Lie symmetries reduction of nonholonomic systems

Abstract

Given a Lie group of finite-dimensional transformations acting on a manifold, there is always an action known as a long-acting group action. This action describes the fundamental basis of the Lie theory connecting groups of symmetry in differential equations. Differential invariants emerge as constants of the action of the prolongation of a group. Élie Cartan extended this in the twentieth century involving the geometry of the action of this group, grounding the so-called moving frame theory. With this theory, various applications are possible and detailed in the literature, such as symmetries of variational problems, conservation laws, invariant differential forms, and group invariant solutions. In order to demonstrate the approach, two nonholonomic constrained mechanical systems are exemplified to obtain either the general closed-solution in explicit form, when possible, or an order reduction provided by the Lie symmetries via moving frames. The first example is a coin with mass m rolling without slipping and takes on an inclined plane (x, y) with angle α and nonlinear constraint. The second example is a Chetaev type described by a dog pursuing a man in a plane surface with a nonholonomic restriction. A full detailed analysis is addressed to define the Lie symmetries and the corresponding moving frames obtained in both examples.