Basic notions of Poisson and symplectic geometry in local coordinates, with applications to Hamiltonian systems

TL;DR

This paper provides an elementary overview of Poisson and symplectic geometries, emphasizing their applications to Hamiltonian systems with constraints, clarifying the Dirac bracket's geometric meaning and its role in system reduction.

Contribution

It offers a clear geometric interpretation of the Dirac bracket and proves the Jacobi identity within Poisson geometry, with applications to constrained Hamiltonian systems.

Findings

Clarified the geometric meaning of the Dirac bracket

Proved the Jacobi identity on a Poisson manifold

Applied Dirac brackets to system reduction and compatibility problems

Abstract

This work contains a brief and elementary exposition of the foundations of Poisson and symplectic geometries, with an emphasis on applications for Hamiltonian systems with second-class constraints. In particular, we clarify the geometric meaning of the Dirac bracket on a symplectic manifold and provide a proof of the Jacobi identity on a Poisson manifold. A number of applications of the Dirac bracket are described: applications for the proof of the compatibility of a system consisting of differential and algebraic equations, as well as applications for the problem of reduction of a Hamiltonian system with known integrals of motion.