Geodesic
The Inverse Problem for Invariant Lagrangians on a Lie Group
Journal of Lie theory, Tome 18 (2008) no. 2, pp. 471-502
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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.
Classification : 22E30, 49N45, 53C22, 53C60, 70H03
Mots-clés : Euler-Poincare equations, inverse problem, Lagrangian system, Lie group, reduction, second-order ordinary differential equations 
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     author = {M. Crampin and T. Mestdag},
     title = {The {Inverse} {Problem} for {Invariant} {Lagrangians} on a {Lie} {Group}},
     journal = {Journal of Lie theory},
     pages = {471--502},
     year = {2008},
     volume = {18},
     number = {2},
     url = {http://geodesic.mathdoc.fr/item/JLT_2008_18_2_JLT_2008_18_2_a13/}
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M. Crampin; T. Mestdag. The Inverse Problem for Invariant Lagrangians on a Lie Group. Journal of Lie theory, Tome 18 (2008) no. 2, pp. 471-502. http://geodesic.mathdoc.fr/item/JLT_2008_18_2_JLT_2008_18_2_a13/