Continuation of Hamiltonian dynamics from the plane to constant-curvature surfaces
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911594183655424 |
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| author | Stoica, Cristina |
| author_facet | Stoica, Cristina |
| contents | 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 $σ\in\{+1,-1\}$, the curved problem is set up either on the sphere $(σ=+1)$ or on the hyperbolic plane $(σ=-1)$, both with radius $R=1/\varepsilon$, recovering flat space in the limit $\varepsilon\to 0$. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_13250 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Continuation of Hamiltonian dynamics from the plane to constant-curvature surfaces Stoica, Cristina Mathematical Physics Dynamical Systems 70Fxx 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 $σ\in\{+1,-1\}$, the curved problem is set up either on the sphere $(σ=+1)$ or on the hyperbolic plane $(σ=-1)$, both with radius $R=1/\varepsilon$, recovering flat space in the limit $\varepsilon\to 0$. 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. |
| title | Continuation of Hamiltonian dynamics from the plane to constant-curvature surfaces |
| topic | Mathematical Physics Dynamical Systems 70Fxx |
| url | https://arxiv.org/abs/2604.13250 |