Group Theoretic Approach to Internal and Collective Degrees of Freedom in Mechanics and Field Theory

Discussed are group-theoretical models of collective degrees of freedom of extended bodies and internal degrees of freedom of point-like objects. We concentrate on the use of groups GL(n, IR), SL(n, IR) and U(n). Relationships with the theory of integrable systems are mentioned

Quantum Systems on Linear Groups

Discussed are quantized dynamical systems on orthogonal and affine groups. The special stress is laid on geodetic systems with affinely-invariant kinetic energy operators. The resulting formulas show that such models may be useful in nuclear and hadronic dynamics. They differ from traditional Bohr-Mottelson models where SL$(n,\mathbb{R})$ is used as a so-called non-invariance group. There is an interesting relationship between classical and quantized integrable lattices

Nonlinear collective nuclear motion

For each real number $\Lambda$ a Lie algebra of nonlinear vector fields on three dimensional Euclidean space is reported. Although each algebra is mathematically isomorphic to $gl(3,{\bf R})$, only the $\Lambda=0$ vector fields correspond to the usual generators of the general linear group. The $\Lambda < 0$ vector fields integrate to a nonstandard action of the general linear group; the $\Lambda >0$ case integrates to a local Lie semigroup. For each $\Lambda$, a family of surfaces is identified that is invariant with respect to the group or semigroup action. For positive $\Lambda$ the surfaces describe fissioning nuclei with a neck, while negative $\Lambda$ surfaces correspond to exotic bubble nuclei. Collective models for neck and bubble nuclei are given by irreducible unitary representations of a fifteen dimensional semidirect sum spectrum generating algebra $gcm(3)$ spanned by its nonlinear $gl(3,{\bf R})$ subalgebra plus an abelian nonlinear inertia tensor subalgebra.Comment: 13 pages plus two figures(available by fax from authors by request

The two-body problem of a pseudo-rigid body and a rigid sphere

In this paper we consider the two-body problem of a spherical pseudo-rigid body and a rigid sphere. Due to the rotational and “re-labelling” symmetries, the system is shown to possess conservation of angular momentum and circulation. We follow a reduction procedure similar to that undertaken in the study of the two-body problem of a rigid body and a sphere so that the computed reduced non-canonical Hamiltonian takes a similar form. We then consider relative equilibria and show that the notions of locally central and planar equilibria coincide. Finally, we show that Riemann’s theorem on pseudo-rigid bodies has an extension to this system for planar relative equilibria