The geometry and mechanics of generalized pseudo-rigid bodies

Abstract

The geometry and the mechanics of generalized pseudo-rigid bodies are studied. The configuration space of a generalized pseudo-rigid body is the linear group, , of non-singular matrices with a positive determinant. It admits the left and right actions of SO(n) and the two-sided action of SO(n) × SO(n). The left and right SO(n) actions on are both free, so that is made into respective principal fiber bundles according to the left and right SO(n) actions, and further left and right connections can be defined on the respective fiber bundles. However, the two-sided SO(n) × SO(n) action is not free on , and hence is not made into a principal fiber bundle with respect to this action. In spite of this, if is restricted to an open dense subset , the isotropy subgroup at each point of is a finite discrete group, so that the quotient space becomes a manifold, and further one can define a connection on , which will be called a bi-connection. The bi-connection is used to reduce the pseudo-rigid body system on with the SO(n) × SO(n) symmetry. Though in the cotangent bundle reduction theorem and its variants, one usually assumes that the action of a Lie group on the configuration space is free, or that the isotropy subgroup of the Lie group is trivial, the reduction procedure works well if the isotropy subgroup is not trivial but a finite group. As an application of the reduction procedure, relative equilibria are discussed in relation with the reduced Hamilton and Lagrange equations of motion. A necessary and sufficient condition is given for a relative equilibrium in terms of an amended potential on the reduced phase space.