  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.
