Nonlinear Effects in a Weakly Nonholonomic Systems With a Small Degrees of Freedom

TL;DR

This paper explores the nonlinear effects in weakly nonholonomic systems with few degrees of freedom, focusing on how small parameters influence system behavior and the transition from holonomic to nonholonomic dynamics.

Contribution

It introduces a normalization approach to analyze nonlinear effects in weakly nonholonomic systems with small degrees of freedom, extending Tatarinov's foundational work.

Findings

Identification of transgression effects in weakly nonholonomic systems

Description of how small parameters modify system dynamics

Extension of integrable Hamiltonian systems to nonholonomic cases

Abstract

In 1986 Ya.V. Tatarinov presented the foundations of the theory of weakly nonholonomic systems. Mechanical systems with nonholonomic constraints depending on a small parameter are considered. It is assumed that for zero value of this parameter the constraints of such a system become integrable; i.e., in this case, we have a family of holonomic systems depending on several arbitrary integration constants. We will assume that these holonomic systems are completely integrable Hamiltonian systems. When the small parameter is not zero, the behavior of such systems can be considered with the help of normalization methods. The behavior of such a system can be represented as a combination of the motion of a slightly modified holonomic system with slowly varying previous integration constants (the transgression effect). In this paper we describe the corresponding effects in a several nonholonomic systems with a small degrees of freedom.