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We extend Routh’s reduction procedure to an arbitrary Lagrangian system (that is, one whose Lagrangian is not necessarily the difference of kinetic and potential energies) with a symmetry group which is not necessarily Abelian. To do so, we analyze the restriction of the Euler–Lagrange field to a level set of momentum in velocity phase space. We present a new method of analysis based on the use of quasivelocities. We discuss the reconstruction of solutions of the full Euler–Lagrange equations from those of the reduced equations. [ABSTRACT FROM AUTHOR]

We introduce the notion of a Lie algebroid structure on an affine bundle whose base manifold is fibered over R. It is argued that this is the framework which one needs for coming to a time-dependent generalization of the theory of Lagrangian systems on Lie algebroids. An extensive discussion is given of a way one can think of forms acting on sections of the affine bundle. It is further shown that the affine Lie algebroid structure gives rise to a coboundary operator on such forms. The concept of admissible curves and dynamical systems whose integral curves are admissible brings an associated affine bundle into the picture, on which one can define in a natural way a prolongation of the original affine Lie algebroid structure. [ABSTRACT FROM AUTHOR]