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A variational framework for accelerated optimization was recently introduced on normed vector spaces and Riemannian manifolds in [1] and [2]. It was observed that a careful combination of time-adaptivity and symplecticity in the numerical integration can result in a significant gain in computational efficiency. It is however well known that symplectic integrators lose their near-energy preservation properties when variable time-steps are used. The most common approach to circumvent this problem involves the Poincaré transformation on the Hamiltonian side, and was used in [3] to construct efficient explicit algorithms for symplectic accelerated optimization. However, the current formulations of Hamiltonian variational integrators do not make intrinsic sense on more general spaces such as Riemannian manifolds and Lie groups. In contrast, Lagrangian variational integrators are well-defined on manifolds, so we develop here a framework for time-adaptivity in Lagrangian variational integrators and use the resulting geometric integrators to solve optimization problems on vector spaces and Lie groups.
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This content will become publicly available on March 1, 2026
Variational principles for Hamiltonian systems
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.
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- PAR ID:
- 10617767
- Publisher / Repository:
- World Scientific
- Date Published:
- Journal Name:
- Geometric Mechanics
- Volume:
- 02
- Issue:
- 01
- ISSN:
- 2972-4589
- Page Range / eLocation ID:
- 59 to 105
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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