Cendra, H., J.E. Marsden and T.S. Ratiu
Reduction theory for mechanical systems with symmetry has its roots in the classical works in mechanics of Euler, Jacobi, Lagrange, Hamilton, Routh, Poincaré and others. The modern vision of mechanics includes, besides the traditional mechanics of particles and rigid bodies, field theories such as electromagnetism, fluid mechanics, plasma physics, solid mechanics as well as quantum mechanics, and relativistic theories, including gravity.
Symmetries in mechanics ranges from obvious translational and rotational symmetries to less obvious particle relabeling symmetries in fluids and plasmas, to subtle symmetries underlying integrable systems. Reduction theory concerns the removal of symmetries and utilizing their associated conservation laws. Reduction theory has been extremely useful in a wide variety of areas, from a deeper understanding of many physical theories, including new variational and Poisson structures, stability theory, integrable systems, as well as geometric phases. Much effort has gone into the development of the symplectic and Poisson view of reduction theory, but recently the Lagrangian view, emphasizing the reduction of variational principles has also matured.
While there has been much activity in the geometry of nonholonomic systems, the task of providing an intrinsic geometric formulation of the reduction theory for nonholonomic systems from the point of view of Lagrangian reduction has been somewhat incomplete. One of the purposes of this paper is to finish this task. In particular, we show how to write the reduced Lagrange d'Alembert equations, and in particular, its vertical part, the momentum equation, intrinsically using covariant derivatives. The resulting equations are called the Lagrange-d'Alembert-Poincaré equations.