
Journal of Lie Theory 18 (2008), No. 2, 471502 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 secondorder 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: EulerPoincare equations, inverse problem, Lagrangian system, Lie group, reduction, secondorder ordinary differential equations. MSC: 22E30, 49N45, 53C22, 53C60, 70H03 [ Fulltextpdf (267 KB)] for subscribers only. 