Actions of the Neumann systems via Picard-Fuchs equations

The Neumann system describing the motion of a particle on an 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 the 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 the related Gauß-Manin equations. These formulas are used for the numerical calculation of the actions of the Neumann system.