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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.