Routh reduction by stages
The process of symmetry reduction for Lagrangian systems is less well-known than the corresponding process for Hamiltonian systems. As with Hamiltonian systems there are two paths that lead to two different Lagrangian reduction theories: Routh and Lagrange-Poincaré reducvtion. This paper addresses the Routh reduction procedure. Routh reduction restricts attention to just those solutions of the system with a prescribed value of momentum. Then the new symmetry group (after reduction) is in general only a subgroup of the fall symmetry group of the original Lagrangian system.NEWLINENEWLINE This paper focuses on those aspects of the theory that are applicable to reduction to stages. When the symmetry group of the system has a normal subgroup, one can first do reduction via this subgroup. One can then ask if there is some residual symmetry, and whether a subsequent second Routh reduction would lead to results that are equivalent to reduction by the full symmetry group.NEWLINENEWLINE The authors identify and develop a category of systems for Routh reduction for which the reduced system at each stage remains in that category. These are the systems they call magnetic Lagrangian systems. The authors also construct a transformation relating the magnetic Lagrangian systems obtained via reduction in stages and the one obtained by reduction via the full symmetry group. This work relies heavily on a generalized version of cotangent symplectic reduction.NEWLINENEWLINE The paper includes several illustrative examples.
scientific article; zbMATH DE number 6050728