| The configuration of a rotationally invariant system is naturally described by its orientation and shape. (Here, shape is the configuration modulo rotations.) Similarly, the velocity of such a system can be decomposed into rotational and vibrational terms. In this dissertation, we address various questions arising from such orientation-shape or rotation-vibration decompositions. We consider the basic structure of shape space and functions defined on it, as well as the coupling of rotations to vibrations and how such coupling is manifest in quantities such as the Hamiltonian. In the final chapter, we develop a general theory of constraints in quantum mechanics. A unifying theme throughout is the use of gauge theory and geometric phase as a natural mathematical tool in both classical and quantum dynamics.;We briefly summarize the specific topics discussed.;We generalize the standard rotation-vibration Hamiltonian for a system of n point particles to a system of n rigid bodies. This generalized Hamiltonian is applicable to molecular clusters.;Using the correspondence between the Coriolis curvature of a three particle system and the magnetic field of a monopole, we unite the theory of three-body hyperspherical harmonics to the well-developed theory of magnetic monopole harmonics.;The wave function Yℓm , defined over configuration space, can be expressed in terms of an internal wave function cℓk , defined over shape space. We prove a set of necessary and sufficient conditions on the internal wave function cℓk to guarantee that Yℓm is smooth anywhere (except the three-body collision).;The kinematic group acts on the n-body shape space, foliating shape space into kinematic orbits. We study the isotropy subgroups and topologies of these kinematic orbits, with a particular emphasis on the five-body problem. The topology of these orbits is crucial for setting up certain molecular computations and is also of intrinsic mathematical interest.;We derive the Hamiltonian for a quantum system confined to a submanifold of configuration space by an infinite restoring force. Surprisingly, the confining potential defines a connection, which appears in the constrained Hamiltonian as a gauge potential. This theory of constrained quantum mechanics has applications to the quantum rigid body, electrons confined to small quantum wires, and chemical reactions. |
