Journal of Lie Theory 18 (2008), No. 2, 471--502
Copyright Heldermann Verlag 2008
The Inverse Problem for Invariant Lagrangians on a Lie Group
Mike Crampin
Dept. of Mathematical Physics and Astronomy, Ghent University, Krijgslaan 281 - S9, 9000 Ghent, Belgium
crampin@btinternet.com
Tom Mestdag
Dept. of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, MI 48109, U.S.A.
and: Dept. of Mathematical Physics and Astronomy, Ghent University, Krijgslaan 281 - S9, 9000 Ghent, Belgium
tom.mestdag@ugent.be
We discuss the problem of the existence of a regular invariant Lagrangian for a given system of invariant second-order ordinary differential equations on a Lie group, using approaches based on the Helmholtz conditions. Although we deal with the problem directly on the tangent manifold of the Lie group, our main result relies on a reduction of the system on the tangent manifold to a system on the Lie algebra of the Lie group. We conclude with some illustrative examples.
Keywords: Euler-Poincare equations, inverse problem, Lagrangian system, Lie group, reduction, second-order ordinary differential equations.
MSC: 22E30, 49N45, 53C22, 53C60, 70H03
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