Continuation of Hamiltonian dynamics from the plane to constant-curvature 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 $R=1/\epsilon$, recovering flat space in the limit $\epsilon \to 0$. The symmetry of these spaces is taken into account by using the Inönü--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.

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