Base-controlled mechanical systems and geometric phases

TL;DR

This paper investigates mechanical systems with controlled base variables, revealing how their motion can be decomposed into dynamical and geometric phases, and providing a reconstruction formula under certain conditions.

Contribution

It introduces a factorization of solutions into dynamical and geometric components for controlled base systems, with a reconstruction phase formula and practical applications.

Findings

Solution factorizes into dynamical and geometric parts

Reconstruction phase formula derived under specific conditions

Applied results to concrete mechanical systems

Abstract

In this paper, we carry a detailed study of mechanical systems with configuration space $Q\longrightarrow Q/G$ for which the base $Q/G$ variables are being controlled. The overall system's motion is considered to be induced from the base one due to the presence of general non-holonomic constraints. It is shown that the solution can be factorized into dynamical and geometrical parts. Moreover, under favorable kinematical circumstances, the dynamical part admits a further factorization since it can be reconstructed from an intermediate (body) momentum solution, yielding a reconstruction phase formula. Finally, we apply this results to the study of concrete mechanical systems.