Dullin, Holger R.; Richter, Peter H.; Veselov, Alexander P.; Waalkens, Holger Actions of the Neumann systems via Picard-Fuchs equations. (English) Zbl 1001.70013 Physica D 155, No. 3-4, 159-183 (2001). 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. Cited in 9 Documents MSC: 70H06 Completely integrable systems and methods of integration for problems in Hamiltonian and Lagrangian mechanics 37N05 Dynamical systems in classical and celestial mechanics Keywords:integrable system; motion of a particle; Neumann system; \(n\)-dimensional sphere; anisotropic harmonic potential; action integrals; Picard-Fuchs equation; Gauß-Manin equations PDFBibTeX XMLCite \textit{H. R. Dullin} et al., Physica D 155, No. 3--4, 159--183 (2001; Zbl 1001.70013) Full Text: DOI