Cotangent bundle reduction and Poincar\'e-Birkhoff normal forms

TL;DR

This paper develops a systematic method for constructing canonical coordinates on cotangent bundle reductions with Lie group symmetries, facilitating the computation of Poincaré-Birkhoff normal forms for relative equilibria in mechanical systems.

Contribution

It introduces a natural construction of canonical coordinates for reduced cotangent bundles, enabling standard normal form calculations for symmetric mechanical systems.

Findings

Computed normal forms for a three-body system with Morse potential.

Analyzed the stretched configuration of a double spherical pendulum.

Demonstrated the method's applicability to systems with Lie group symmetries.

Abstract

In this paper we study a systematic and natural construction of canonical coordinates for the reduced space of a cotangent bundle with a free Lie group action. The canonical coordinates enable us to compute Poincar{\'e}-Birkhoff normal forms of relative equilibria using standard algorithms. The case of simple mechanical systems with symmetries is studied in detail. As examples we compute Poincar{\'e}-Birkhoff normal forms for a Lagrangian equilateral triangle configuration of a three-body system with a Morse-type potential and the stretched-out configuration of a double spherical pendulum.