Abelian integrals and the reduction method for an integrable Hamiltonian system

Abstract

A classical finite-dimensional integrable Hamiltonian system, corresponding to the motion of a particle constrained to an n-dimensional sphere ∑nμ=0 x2μ=1, with the Hamiltonian H=∑μ( (1)/(2) y2μ+u2μ/x2μ +εαμx2μ) (where uμ, αμ, and ε are constants and yμ are the momenta conjugate to xμ), is integrated using several different methods. These are the following: (1) The projection of geodesic (free) flow on a larger space, namely the sphere S2n+1 (for ε=0). The flow is obtained in terms of elementary functions. (2) Separation of variables in the Hamilton–Jacobi equation in elliptic coordinates or, alternatively, the use of a complete set of integrals of motion in involution to reduce Hamilton’s equations to quadratures. The flow is obtained in terms of Abelian integrals which are then inverted in terms of generalized θ functions. The relation between the different methods and results is clarified using methods of algebraic geometry, in particular the geometry of quadrics.