Symplectic manifolds and Hamiltonian dynamical systems

TL;DR

This paper explores the properties of symplectic manifolds and their role in Hamiltonian systems, providing detailed proofs, explicit calculations, and applications to demonstrate their significance in integrability and dynamics.

Contribution

It offers a comprehensive review of symplectic manifolds, detailed proofs, and explicit calculations, linking these structures to Hamiltonian system integrability and dynamics.

Findings

Properties and operations on symplectic manifolds are elucidated.

Connections between symplectic geometry and Hamiltonian integrability are established.

Explicit calculations and applications to dynamical systems are provided.

Abstract

This paper is devoted to the study of symplectic manifolds and their connection with Hamiltonian dynamical systems. We review some properties and operations on these manifolds and see how they intervene when studying the complete integrability of these systems, with detailed proofs. Several explicit calculations for which references are not immediately available are given. These results are exemplified by applications to some Hamiltonian dynamical systems.