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Actions of the Neumann systems via Picard-Fuchs equations. (English) Zbl 1001.70013

Summary: The Neumann system describing the motion of a particle on \(n\)-dimensional sphere with an anisotropic harmonic potential has been celebrated as one of the best understood integrable systems of classical mechanics. The present paper adds a detailed discussion and the determination of its action integrals, using differential equations rather than standard integral formulas. We show that the actions of Neumann system satisfy a Picard-Fuchs equation, which in suitable coordinates has a rather simple form for arbitrary \(n\). We also present an explicit form of related Gauß-Manin equations. These formulas are used for numerical calculation of the actions of Neumann system.

MSC:

70H06 Completely integrable systems and methods of integration for problems in Hamiltonian and Lagrangian mechanics
37N05 Dynamical systems in classical and celestial mechanics
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