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Geometry of Mechanics

Miguel C. Muñoz-Lecanda, Narciso Román-Roy

TL;DR

This work provides a comprehensive geometric tour of mechanics, detailing autonomous and nonautonomous, as well as dissipative systems, through symplectic, cosymplectic, and contact formalisms. It builds from foundational symplectic geometry to Lagrangian and canonical Hamiltonian descriptions, then unifies them via Skinner–Rusk’s framework, all while addressing symmetries, conserved quantities, and reduction. Central to the exposition are momentum maps, Noether’s theorem, and Hamilton–Jacobi theory, which together yield a cohesive toolbox for analyzing regular and, when possible, singular dynamics. The treatment emphasizes both local (coordinate) expressions and global geometric structures, enabling rigorous analysis of classical problems and laying groundwork for extensions to more general and dissipative contexts. Overall, the geometry of mechanics emerges as a versatile, unifying language for describing the dynamics of physical systems through canonical forms, variational principles, and symmetry-driven reductions.

Abstract

The aim of this work is to study the geometry underlying mechanics and its application to describe autonomous and nonautonomous conservative dynamical systems of different types; as well as dissipative dynamical systems. We use different geometric descriptions to study the main properties and characteristics of these systems; such as their Lagrangian, Hamiltonian and unified formalisms, their symmetries, the variational principles, and others. The study is done mainly for the regular case, although some comments and explanations about singular systems are also included.

Geometry of Mechanics

TL;DR

This work provides a comprehensive geometric tour of mechanics, detailing autonomous and nonautonomous, as well as dissipative systems, through symplectic, cosymplectic, and contact formalisms. It builds from foundational symplectic geometry to Lagrangian and canonical Hamiltonian descriptions, then unifies them via Skinner–Rusk’s framework, all while addressing symmetries, conserved quantities, and reduction. Central to the exposition are momentum maps, Noether’s theorem, and Hamilton–Jacobi theory, which together yield a cohesive toolbox for analyzing regular and, when possible, singular dynamics. The treatment emphasizes both local (coordinate) expressions and global geometric structures, enabling rigorous analysis of classical problems and laying groundwork for extensions to more general and dissipative contexts. Overall, the geometry of mechanics emerges as a versatile, unifying language for describing the dynamics of physical systems through canonical forms, variational principles, and symmetry-driven reductions.

Abstract

The aim of this work is to study the geometry underlying mechanics and its application to describe autonomous and nonautonomous conservative dynamical systems of different types; as well as dissipative dynamical systems. We use different geometric descriptions to study the main properties and characteristics of these systems; such as their Lagrangian, Hamiltonian and unified formalisms, their symmetries, the variational principles, and others. The study is done mainly for the regular case, although some comments and explanations about singular systems are also included.
Paper Structure (155 sections, 145 theorems, 725 equations)

This paper contains 155 sections, 145 theorems, 725 equations.

Table of Contents

  1. Introduction
  2. Symplectic mechanics (I): Autonomous Hamiltonian dynamical systems
  3. Notions on symplectic and presymplectic geometry
  4. Symplectic and presymplectic vector spaces
  5. Subspaces of a symplectic linear space
  6. Symplectic and presymplectic manifolds
  7. Submanifolds of a symplectic manifold
  8. Canonical isomorphism. Hamiltonian vector fields
  9. Invariant forms
  10. Poisson brackets
  11. Canonical transformations and symplectomorphisms
  12. Characterization of canonical transformations
  13. Generating functions of canonical transformations
  14. Hamiltonian dynamical systems
  15. Hamiltonian systems
  16. ...and 140 more sections

Key Result

Lemma 2.2

Let $({\bf E},\omega)$ be a $2n$ dimensional symplectic vector space.

Theorems & Definitions (473)

  • Definition 2.1
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Definition 2.4
  • Proposition 2.5
  • proof
  • Proposition 2.6
  • proof
  • Definition 2.7
  • ...and 463 more