Lie-Hamilton systems on curved spaces: A geometrical approach

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.

arXiv comment20 pages. Some misprints corrected. Some comments added

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• Geometric features of Vessiot--Guldberg Lie algebras of conformal and Killing vector fields on ${\mathbb{R}}^2$
• Lie-Hamilton systems on Riemannian and Lorentzian spaces from conformal transformations and some of their applications
• Contact Lie systems on Riemannian and Lorentzian spaces: from scaling symmetries to curvature-dependent reductions
• Nonlinear Lie-Hamilton systems: $t$-Dependent curved oscillators and Kepler-Coulomb Hamiltonians
• Conformal Parametrisation of Loxodromes by Triples of Circles
• Geometric numerical methods for Lie systems and their application in optimal control
• Reduction and reconstruction of multisymplectic Lie systems
• Geometry preserving numerical methods for physical systems with finite-dimensional Lie algebras