Lie-Hamilton systems on curved spaces: A geometrical approach

TL;DR

This paper explores Lie-Hamilton systems on various curved and semi-Riemannian spaces, providing explicit geometric methods to find constants of motion and superposition rules, thus extending algebraic contraction techniques.

Contribution

It introduces a geometric approach to Lie-Hamilton systems on diverse spaces, extending algebraic contraction methods to vector fields, Hamiltonian functions, and symplectic structures.

Findings

Explicit constants of motion derived for systems on curved spaces

Superposition rules formulated geometrically for various geometries

Extension of algebraic contraction to geometric structures

Abstract

A Lie-Hamilton system is a nonautonomous system of first-order ordinary differential equations describing the integral curves of a $t$-dependent vector field taking values in a finite-dimensional Lie algebra, a Vessiot-Guldberg Lie algebra, of Hamiltonian vector fields relative to a Poisson structure. Its general solution can be written as an autonomous function, the superposition rule, of a generic finite family of particular solutions and a set of constants. We pioneer the study of Lie-Hamilton systems on Riemannian spaces (sphere, Euclidean and hyperbolic plane), pseudo-Riemannian spaces (anti-de Sitter, de Sitter, and Minkowski spacetimes) as well as on semi-Riemannian spaces (Newtonian spacetimes). Their corresponding constants of motion and superposition rules are obtained explicitly in a geometric way. This work extends the (graded) contraction of Lie algebras to a contraction procedure for Lie algebras of vector fields, Hamiltonian functions, and related symplectic structures, invariants, and superposition rules.