A Type II Hamiltonian Variational Principle and Adjoint Systems for Lie Groups
Brian K. Tran, Melvin Leok
TL;DR
This work develops a global Type II Hamiltonian variational principle on the cotangent bundle $T^*G$ of a Lie group, using left-trivialization to impose fixed initial position and final momentum boundary conditions and derive the corresponding Hamiltonian and Lie-Poisson dynamics. It then specializes to adjoint systems for dynamics on $G$, establishing a global existence/uniqueness theory and a quadratic conservation law that underpins adjoint sensitivity analysis. The framework is extended to discrete dynamics via Lie group variational integrators that preserve symplectic structure and Noether properties, and a discrete adjoint sensitivity methodology is derived that yields exact discrete gradients for terminal costs. The results enable structure-preserving sensitivity analysis and optimization for systems evolving on Lie groups, with potential impact on geometric control, autonomous design, and symmetry-aware neural networks.
Abstract
We present a novel Type II variational principle on the cotangent bundle of a Lie group which enforces Type II boundary conditions, i.e., fixed initial position and final momentum. In general, such Type II variational principles are only globally defined on vector spaces or locally defined on general manifolds; however, by left translation, we are able to define this variational principle globally on cotangent bundles of Lie groups. Type II boundary conditions are particularly important for adjoint sensitivity analysis, which is our motivating application. As such, we additionally discuss adjoint systems on Lie groups, their properties, and how they can be used to solve optimization problems subject to dynamics on Lie groups.