2026/04/14 by Cristina Stoica
Mathematics · #math-ph #math.DS #math.MP
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 σ∈\+1,-1\, the curved problem is set up either on the sphere (σ=+1) or on the hyperbolic plane (σ=-1), both with radius R=1/ε, recovering flat space in the limit ε→ 0. The symmetry of these spaces is taken into account by using the Inönü--Wigner contraction of Lie algebras from \mathfrakso(3) or \mathfrakso(2,1) to \mathfrakse(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.