Preservation of Symmetry in Hamiltonian Systems: A Study on Simulation and Learning

[Submitted on 30 Aug 2023]

Download a PDF of the paper titled “Symmetry Preservation in Hamiltonian Systems: Simulation and Learning” by Miguel Vaquero and 1 other author.

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Abstract: This work introduces a comprehensive geometric framework for simulating and learning the dynamics of Hamiltonian systems that possess invariance under a Lie group of transformations. These systems exhibit symmetries that respect their dynamics, resulting in the observation of conserved quantities as per Noether’s Theorem. Our approach involves simulating and learning the relevant mappings through the construction of G-invariant Lagrangian submanifolds, which are crucial objects in symplectic geometry. Notably, our constructions ensure that the simulated/learned dynamics also preserve the same conserved quantities as the original system, thereby providing a more accurate representation of the original dynamics compared to non-symmetry aware methods. Additionally, our framework can simulate/learn not only Hamiltonian flows but also any Lie group-equivariant symplectic transformation. Our designs leverage important techniques and concepts in symplectic geometry and geometric mechanics, including reduction theory, Noether’s Theorem, Lagrangian submanifolds, momentum mappings, and coisotropic reduction, among others. We also present methods to learn Poisson transformations while preserving the underlying geometry and how to imbue non-geometric integrators with geometric properties. Thus, this work presents an innovative endeavor to harness the power of symplectic and Poisson geometry in simulating and learning problems.