Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 7 (2011), 109, 31 pages      arXiv:1106.2950      https://doi.org/10.3842/SIGMA.2011.109

Routh Reduction by Stages

Bavo Langerock a, b, c, Tom Mestdag a and Joris Vankerschaver a, d
a) Department of Mathematics, Ghent University, Krijgslaan 281, S22, B9000 Ghent, Belgium
b) Belgian Institute for Space Aeronomy, Ringlaan 3, B1180 Brussels, Belgium
c) Department of Mathematics, K.U. Leuven, Celestijnenlaan 200 B, B3001 Leuven, Belgium
d) Department of Mathematics, University of California at San Diego, 9500 Gilman Drive, San Diego CA 92093-0112, USA

Received June 16, 2011, in final form November 22, 2011; Published online November 29, 2011

Abstract
This paper deals with the Lagrangian analogue of symplectic or point reduction by stages. We develop Routh reduction as a reduction technique that preserves the Lagrangian nature of the dynamics. To do so we heavily rely on the relation between Routh reduction and cotangent symplectic reduction. The main results in this paper are: (i) we develop a class of so called magnetic Lagrangian systems and this class has the property that it is closed under Routh reduction; (ii) we construct a transformation relating the magnetic Lagrangian system obtained after two subsequent Routh reductions and the magnetic Lagrangian system obtained after Routh reduction w.r.t. to the full symmetry group.

Key words: symplectic reduction; Routh reduction; Lagrangian reduction; reduction by stages.

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