Continuation of Hamiltonian dynamics from the plane to constant-curvature surfaces

TL;DR

This paper explores how Hamiltonian dynamics and symmetries deform when transitioning from flat Euclidean space to curved surfaces like spheres and hyperbolic planes, using geometric and algebraic methods.

Contribution

It develops a framework for understanding the persistence of relative equilibria and periodic orbits in Hamiltonian systems on curved surfaces, extending flat space results.

Findings

Persistence of non-degenerate relative equilibria from flat to curved spaces

Persistence of relative periodic orbits under curvature deformation

Application to the Newtonian n-body problem on curved surfaces

Abstract

We investigate the deformation of symmetry on cotangent bundles from the Euclidean plane to two-dimensional constant-curvature surfaces and the continuation of local dynamics aspects in Hamiltonian systems. For a fixed curvature sign $\sigma\in\{+1,-1\}$, the curved problem is set up either on the sphere $(\sigma=+1)$ or on the hyperbolic plane $(\sigma=-1)$, both with radius $R=1/\varepsilon$, recovering flat space in the limit $\varepsilon\to 0$. The symmetry of these spaces is taken into account by using the In\"on\"u--Wigner contraction of Lie algebras from $\mathfrak{so}(3)$ or $\mathfrak{so}(2,1)$ to $\mathfrak{se}(2)$. We use Riemannian exponential coordinates centred at the North pole together with the pull-back the associated momentum map and the symplectic form. Within this geometric setting we use a local slice construction and prove the persistence from flat to curved spaces of non-degenerate relative equilibria and relative periodic orbits of general cotangent bundle Hamiltonian systems. We apply the resulting framework to the Newtonian $n$-body problem.