Mechanical Systems : Symmetries and Reduction

Abstract

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. Momentummapping 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. Equivariance Equivariance of a momentum map is a property that reflects the consistency of the mapping with a group action on its domain and range. Momentum cocycle A measure of the lack of equivariance of a momentummapping. Singular reduction A reduction process that leads to non-smooth reduced spaces. Often associated with non-free group actions. Coadjoint orbit The orbit of an element of the dual of the Lie algebra under the natural action of the group. KKS (Kostant-Kirillov-Souriau) form The natural symplectic form on coadjoint orbits. Cotangent bundle A mechanical phase space that has a structure that distinguishes configurations and momenta. The momenta lie in the dual to the space of velocity vectors of configurations. Shape space The space obtained by taking the quotient of the configuration space of a mechanical system by the symmetry group. Principal connection A mathematical object that describes the geometry of how a configuration space is related to its shape space. Related to geometric phases through the subject of holonomy. In turn related to locomotion in mechanical systems. Mechanical connection A special (principal) connection that is built out of the kinetic energy and momentum map of a mechanical system with symmetry. Magnetic terms These are expressions that are built out of the curvature of a connection. They are so named because terms of this form occur in the equations of a particle moving in a magnetic field.