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Variational Principles for Hamiltonian Systems

Brian K. Tran, Melvin Leok

TL;DR

This work introduces new Type II variational principles for Hamiltonian systems based on a virtual work approach, enabling intrinsic formulations on parallelizable and general manifolds and extending to infinite-dimensional settings. It analyzes boundary conditions, develops a d'Alembert-type variant, and reformulates the theory as a free boundary problem to achieve global applicability. The authors survey and connect these principles to Hamiltonian variational integrators, adjoint sensitivity analysis, optimal control, Hamiltonian PDEs, constrained systems, and stochastic Hamiltonian dynamics, highlighting both theoretical and practical implications. The framework lays the groundwork for intrinsic, manifold-aware variational methods and integrators applicable to forward and adjoint problems across finite and infinite dimensions.

Abstract

Motivated by recent developments in Hamiltonian variational principles, Hamiltonian variational integrators, and their applications such as to optimization and control, we present a new Type II variational approach for Hamiltonian systems, based on a virtual work principle that enforces the Type II boundary conditions through a combination of essential and natural boundary conditions; particularly, this approach allows us to define this variational principle intrinsically on manifolds. We first develop this variational principle on vector spaces and subsequently extend it to parallelizable manifolds, general manifolds, as well as to the infinite-dimensional setting. Furthermore, we provide a review of variational principles for Hamiltonian systems in various settings as well as their applications.

Variational Principles for Hamiltonian Systems

TL;DR

This work introduces new Type II variational principles for Hamiltonian systems based on a virtual work approach, enabling intrinsic formulations on parallelizable and general manifolds and extending to infinite-dimensional settings. It analyzes boundary conditions, develops a d'Alembert-type variant, and reformulates the theory as a free boundary problem to achieve global applicability. The authors survey and connect these principles to Hamiltonian variational integrators, adjoint sensitivity analysis, optimal control, Hamiltonian PDEs, constrained systems, and stochastic Hamiltonian dynamics, highlighting both theoretical and practical implications. The framework lays the groundwork for intrinsic, manifold-aware variational methods and integrators applicable to forward and adjoint problems across finite and infinite dimensions.

Abstract

Motivated by recent developments in Hamiltonian variational principles, Hamiltonian variational integrators, and their applications such as to optimization and control, we present a new Type II variational approach for Hamiltonian systems, based on a virtual work principle that enforces the Type II boundary conditions through a combination of essential and natural boundary conditions; particularly, this approach allows us to define this variational principle intrinsically on manifolds. We first develop this variational principle on vector spaces and subsequently extend it to parallelizable manifolds, general manifolds, as well as to the infinite-dimensional setting. Furthermore, we provide a review of variational principles for Hamiltonian systems in various settings as well as their applications.
Paper Structure (20 sections, 10 theorems, 135 equations, 1 figure, 2 tables)

This paper contains 20 sections, 10 theorems, 135 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Lagrangian and Hamiltonian Mechanics
  3. Hamiltonian Mechanics from the Type II/III Perspective
  4. Boundary Conditions
  5. Type II/III Variational Principles
  6. Type II Variational Principle on Vector Spaces
  7. Type II d'Alembert Variational Principle on Vector Spaces
  8. Type II Variational Principle for Parallelizable Manifolds
  9. Global Type II Variational Principle as a Free Boundary Problem
  10. Infinite-dimensional Type II Variational Principles
  11. Overview of Hamiltonian Variational Principles
  12. Lagrangian and Hamiltonian Variational Integrators
  13. Optimization and Control
  14. Adjoint Sensitivity Analysis
  15. Optimal Control
  16. ...and 5 more sections

Key Result

Proposition 1.1

Hamilton's variational principle holds if and only if the Euler--Lagrange equation with Type I boundary conditions holds:

Figures (1)

  • Figure 1: Schematic of Type I variations (left) and partial variations (right) on the base manifold.

Theorems & Definitions (25)

  • Proposition 1.1
  • Proposition 1.2
  • Definition 2.1: Regularity and Degeneracy
  • Remark 2.1
  • Remark 2.2: Maximally Degenerate Hamiltonians
  • Proposition 2.1: Type II Variational Principle on a Vector Space
  • proof
  • Remark 2.3
  • Proposition 2.2: Type II d'Alembert Variational Principle
  • proof
  • ...and 15 more