Reduction theory and the Lagrange-Routh equations

Reduction theory for mechanical systems with symmetry has its roots in the classical works in mechanics of Euler, Jacobi, Lagrange, Hamilton, Routh, P oincare, and others. The modern vision of mechanics includes, besides the t raditional 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. Symmetr ies in these theories vary from obvious translational and rotational symmet ries to less obvious particle relabeling symmetries in fluids and plasmas, to subtle symmetries underlying integrable systems. Reduction theory concer ns the removal of symmetries and their associated conservation laws. Variat ional principles, along with symplectic and Poisson geometry, provide funda mental tools for this endeavor. Reduction theory has been extremely useful in a wide variety of areas, from a deeper understanding of many physical th eories, including new variational and Poisson structures, to stability theo ry, integrable systems, as well as geometric phases. This paper surveys pro gress in selected topics in reduction theory, especially those of the last few decades as well as presenting new results on non-Abelian Routh reductio n. We develop the geometry of the associated Lagrange-Routh equations in so me detail. The paper puts the new results in the general context of reducti on theory and discusses some future directions. (C) 2000 American Institute of Physics. [S0022-2488(00)02006-5].