Invariant reduction of the two-body problem with central interaction on simply connected spaces of constant sectional curvature. (English) Zbl 1014.70009

Shchepetilov, Alexey V.

Mladenov, I. M. (ed.) et al., Proceedings of the international conference on geometry, integrability and quantization, Varna, Bulgaria, September 1-10, 1999. Sofia: Coral Press Scientific Publishing. 229-240 (2000).

The author considers two classical particles with central intraction on simply connected spaces of constant curvature from the invariant point of view. He uses Hamiltonian reduction method for excluding a motion of the system as a whole. By using this reduction, the author mentions that the classical two-body problem on the sphere and on the hyperbolic space reaches its maximal generality for three-dimensional spaces. So, he considers the two-body motion on three-dimensional constant curvature spaces. In section 2, the author prepares basic notations used below. In section 3, phase spaces and Hamilton functions are discussed for the sphere and for the hyperbolic space. From the invariant point of view, in section 4, the author derives a result on the Hamiltonian reduction. The final sections 5 and 6 discuss reduction of two-body system on the sphere and on the hyperbolic space.
For the entire collection see [Zbl 0940.00039].

Reviewer: T.Nono (Hiroshima)

MSC:

70F05	Two-body problems
70G45	Differential geometric methods (tensors, connections, symplectic, Poisson, contact, Riemannian, nonholonomic, etc.) for problems in mechanics
70H33	Symmetries and conservation laws, reverse symmetries, invariant manifolds and their bifurcations, reduction for problems in Hamiltonian and Lagrangian mechanics
53Z05	Applications of differential geometry to physics

Keywords:

central intraction; Hamiltonian reduction method; two-body problem; sphere; hyperbolic space; three-dimensional constant curvature spaces; phase spaces