Symmetries and Reduction -- part I -- Poisson and symplectic picture

TL;DR

This paper explores the reduction of classical dynamics using Poisson and symplectic formalisms, emphasizing the role of symmetries and Noether's theorem in simplifying complex systems.

Contribution

It introduces a general framework for reduction in classical mechanics focusing on Poisson and symplectic structures, highlighting the connection with symmetries and conservation laws.

Findings

Framework for reduction in classical dynamics established

Role of Noether's theorem clarified in the reduction process

Focus on Poisson and symplectic formalisms

Abstract

Coherently with the principle of analogy suggested by Dirac, we describe a general setting for reducing a classical dynamics, and the role of the Noether theorem -- connecting symmetries with constants of the motion -- within a reduction. This is the first of two papers, and it focuses on the reduction within the Poisson and the symplectic formalism.