Lagrangian reduction by stages for non-holonomic systems in a Lie algebroid framework

The Lagrange-d’Alembert equations of a non-holonomic system with symmetry can be reduced to the Lagrange-d’Alembert-Poincaré equations. In a previous contribution we have shown that both sets of equations fall in the category of so-called ‘Lagrangian systems on a subbundle of a Lie algebroid’. In this paper, we investigate the special case when the reduced system is again invariant under a new symmetry group (and so forth). Via Lie algebroid theory, we develop a geometric context in which successive reduction can be performed in an intrinsic way. We prove that, at each stage of the reduction, the reduced systems are part of the above mentioned category, and that the Lie algebroid structure in each new step is the quotient Lie algebroid of the previous step. We further show that that reduction in two stages is equivalent with direct reduction. PACS numbers: 02.40.-k, 45.20.Jj