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 these theories vary 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 their associated conservation laws. Variational principles along with symplectic and Poisson geometry, provide fundamental tools for this endeavor. 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
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