Mechanical Systems: Symmetry and Reduction

Lie Group Action. A process by which a Lie group, acting as a symmetry, moves points in a space. When points in the space that are related by a group element are identified, one obtains the quotient space. Free Action. An action that moves every point under any nontrivial group element. Proper Action. An action that obeys a compactness condition. Momentum Mapping. A dynamically conserved quantity that is associated with the symmetry of a mechanical system. An example is angular momentum, which is associated with rotational symmetry. Symplectic Reduction. A process of reducing the dimension of the phase space of a mechanical system by restricting to the level set of a momentum map and also identifying phase space points that are related by a symmetry. Poisson Reduction. A process of reducing the dimension of the phase space of a mechanical system by identifying phase space points that are related by a symmetry.