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Download file# Actions of the Neumann systems via Picard-Fuchs equations

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posted on 06.02.2006, 16:29 by Holger R. Dullin, Peter H. Richter, Alexander VeselovAlexander Veselov, Holger WaalkensThe 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.

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