Abstract
The mathematical models underlying physics and engineering applications are often limited by constraints. In mechanics, when the constraints are velocity-dependent (or: limiting the admissible directions), and when they can not be integrated to a merely position-depending form, they are called nonholonomic constraints. In this project we will investigate the almost-symplectic and almost- Poisson geometry behind Lagrangian systems with such constraints. In particular, we focus on the nonholonomic exponential map for general Lagrangian functions and on the curvature of nonlinear nonholonomic constraints.
Chaplygin systems are a subclass of nonholonomic systems with a symmetry group. We first make for this class an in-depth study of the nonholonomic exponential map, in the case of a kinetic energy Lagrangian. Second, we use techniques from Finsler geometry to extend the results to the situation of a general Lagrangian function. Next, we consider nonlinear nonholonomic constraints. They can be represented by the horizontal manifold of a nonlinear splitting on a fibre bundle. The concept of a nonlinear splitting extends that of an Ehresmann connection. We examine the role of its curvature, in the context of nonholonomic systems.
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