Symplectic geometry [SMATB307]

Objectives

Introduction to Hamiltonian mechanics : Hamilton equations, canonical transformations, Hamilton-Jacobi equation, action-angle variables.

Symplectic geometry is at the foundations of Hamiltonian theory; one can go even further by saying that « Hamiltonian mechanics is a geometry on a symplectic manifold ». Starting from simple examples, we show the link existing between geometry and classical mechanics (Lagrangian and Hamiltonian). We recall the basic notions of differential geometry (manifold, tangent space) and introduce the notion of differential forms, to finally define a symplectic manifold. We introduce the notion of Hamiltonian function. We study the properties of Hamiltonian systems under canonical changes of variables and show how to determine if a coordinate transformation is canonical and how to build them. The final lectures consist in the construction of action-angle variables and an application to the 2-body problem.

Table of contents

Chapter I. Lagrangian mechanics - Chapitre II. Introduction to Hamiltonian mechanics - Chapitre III. Canonical formalism - Chapitre IV. Hamilton-Jacobi theory and Action-Angle variables - Chapitre V. The 2-body problem.