Variational Principles for Hamiltonian Systems

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.

To appear in: Geometric Mechanics

References in corpus (13)

• Stochastic Variational Integrators
• Geometric Computational Electrodynamics with Variational Integrators and Discrete Differential Forms
• Variational integration for ideal magnetohydrodynamics with built-in advection equations
• Stochastic Discrete Hamiltonian Variational Integrators
• Variational and Geometric Structures of Discrete Dirac Mechanics
• Discrete Variational Optimal Control
• Multisymplecticity of hybridizable discontinuous Galerkin methods
• Generating Functionals and Lagrangian PDEs
• Hamilton-Jacobi theory in multisymplectic classical field theories
• Practical Perspectives on Symplectic Accelerated Optimization
• Time-adaptive Lagrangian Variational Integrators for Accelerated Optimization on Manifolds
• A comparison of vakonomic and nonholonomic dynamics with applications to non-invariant Chaplygin systems
• Boundary Conditions for Constraint Systems in Variational Principle