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.