Geometric Models for Lie–Hamilton Systems on ℝ2

Abstract

This paper provides a geometric description for Lie-Hamilton systems on R 2 with locally transitive Vessiot-Guldberg Lie algebras through two types of geometric models. The first one is the restriction of a class of Lie-Hamilton systems on the dual of a Lie algebra to even-dimensional symplectic leaves relative to the Kirillov-Kostant-Souriau bracket. The second is a projection onto a quotient space of an automorphic Lie-Hamilton system relative to a naturally defined Poisson structure or, more generally, an automorphic Lie system with a compatible bivector field. These models give rise to a natural framework for the analysis of Lie-Hamilton systems on R 2 while retrieving known results in a natural manner. Our methods may be extended to study Lie-Hamilton systems on higher-dimensional manifolds and provide new approaches to Lie systems admitting compatible geometric structures.

Geometric Numerical Methods for Lie Systems and Their Application in Optimal Control

“…In addition, since the exponential is a local diffeomorphism, the topological study of matrix Lie groups would allow us to establish the optimal time-step for Lie group methods, which is a long-standing problem that would also help optimize Lie system methods. Another endeavor is to study Lie systems on more general manifolds that are not necessarily isomorphic to R n and depict how some geometric and topological invariants are preserved [50,51]. Right now, we are working on examples on Anti-de-Sitter spaces so we can depict how the curvature is preserved under the numerical method.…”

Geometric numerical methods for Lie systems and their application in optimal control

“…In addition, since the exponential is a local diffeomorphism, the topological study of matrix Lie groups would allow us to establish the optimal time-step for Lie group methods, which is a long-standing problem that would also help optimize Lie system methods. We will also study Lie systems a manifold N different of R n to understand features due to the topology of N [19,29].…”