2134/759 Holger R. Dullin Holger R. Dullin Peter H. Richter Peter H. Richter Alexander Veselov Alexander Veselov Holger Waalkens Holger Waalkens Actions of the Neumann systems via Picard-Fuchs equations Loughborough University 2006 untagged Mathematical Sciences not elsewhere classified 2006-02-06 16:29:20 Preprint https://repository.lboro.ac.uk/articles/preprint/Actions_of_the_Neumann_systems_via_Picard-Fuchs_equations/9381014 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.