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Abelian integrals and the reduction method for an integrable Hamiltonian system. (English) Zbl 0597.70020
(Authors’ summary.) A classical finite-dimensional integrable Hamiltonian system, corresponding to the motion of a particle constrained to an n- dimensional sphere \(\sum^{n}_{\mu =0}x^ 2_{\mu}=1\), with the Hamiltonian \(H=\sum_{\mu}(y^ 2_{\mu}+u^ 2_{\mu}/x^ 2_{\mu}+\epsilon \alpha_{\mu}x^ 2_{\mu})\) (where \(u_{\mu}\), \(\alpha_{\mu}\), and \(\epsilon\) are constants and \(y_{\mu}\) are the momenta conjugate to \(x_{\mu})\), is integrated using several different methods. These are the following:
(1) The projection of geodesic (free) flow on a larger space, namely the sphere \(S^{2n+1}\) (for \(\epsilon =0)\). 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 \(\theta\) functions. The relation between the different methods and results is clarified using methods of algebraic geometry, in particular the geometry of quadrics.
Reviewer: D.Edelen

MSC:
70H20 Hamilton-Jacobi equations in mechanics
70H05 Hamilton’s equations
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
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