Reduction theory for mechanical systems with symmetry has its roots in the classical works in mechanics of Euler, Jacobi, Lagrange, Hamilton, Routh, Poincar´e 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.
This paper surveys progress in selected topics in reduction theory, especially those
of the last few decades as well as presenting new results on nonabelian Routh reduction.
We develop the geometry of the associated Lagrange–Routh equations in some detail.
The paper puts the new results in the general context of reduction theory and discusses
some future directions.