Journal of Lie Theory Vol. 14, No. 2, pp. 395425 (2004) 

Variationality of fourdimensional Lie group connectionsR. Ghanam, G. Thompson, and E. J. Miller%Ryad Ghanam Department of Mathematics University of Pittsburgh at Greensburg Greensburg, PA 15601, USA ghanam@pitt.edu and G. Thompson, and E.J. Miller Department of Mathematics University of Toledo Toledo, OH 43606, USA gthomps@uoft02.utoledo.edu Abstract: Following on from previous work by one of the authors on dimensions two and three, this paper gives a comprehensive analysis of the inverse problem of Lagrangian dynamics for the geodesic equations of the canonical linear connection on Lie groups of dimension four. Starting from the Lie algebra, in every case a faithful fourdimensional representation of the algebra is given as well as one in terms of vector fields and a representation of the linear group of which the given algebra is its Lie algebra. In each case the geodesic equations are calculated as a starting point for the inverse problem. Some results about first integrals of the geodesics are obtained. It is found that in three classes of algebra, there are algebraic obstructions to the existence of a Lagrangian, which can be determined directly from the Lie algebra without the need for any representation. In all other cases there are Lagrangians and indeed whole families of them. In many cases a formula for the most general Hessian of a Lagrangian is obtained. \hfill\break {\eightsl AMS Subject classification}: 70H30,70H06,70H03,53B40,53C60,57S25. \hfill\break {\eightsl Key Words}: canonical symmetric connection, Lie group, Lie algebra, EulerLagrange equations, Lagrangian, first integral of geodesic Full text of the article:
Electronic version published on: 1 Sep 2004. This page was last modified: 1 Sep 2004.
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