Institute for Fundamental Technological Problems,

Polish Academy of Sciences,

21 Swietokrzyska Str., 00-049 Warsaw, POLAND

E-mail: vkoval@ippt.gov.pl

**Affine models of collective and internal degrees of
freedom. Problems of dynamical affine invariance**

**Abstract:**

The linear group GL(3, **R**) and its unimodular subgroup SL(3,**R**)
have been used for a long time in various physical problems, e.g. in nuclear
physics, theory of molecular vibrations, macroscopic elasticity and even
in astrophysics. In all these problems geometry of degrees of freedom was
ruled by the linear or affine groups, but the dynamics itself was invariant
under smaller groups, usually SO(3,*R*). In nuclear physics SL(3,**R**)
was used as a dynamical non-invariance group.

We present some models where the kinetic energy itself is also invariant
under SL(3,**R**). So, without potential, we use geodetic models on
SL(3,**R**) invariant under this group or even under the direct product
SL(3,**R**) x SL(3,**R**). For mathematical generality we consider
the *n*-dimensional case SL(*n*,**R**). It turns out that
in spite of the non-compact character of this group, there exists an open
family of bounded classical motions. On the quantum level this is expressed
by the existence of discrete energy spectra. One can also consider more
general case, when some potential term is also admitted (at least the one
stabilizing the volume object). On the classical level the problem is analyzed
with the use of appropriate Poisson structures.

There are some reasons to expect that such models may be physically applicable in nuclear physics, astrophysics and even in certain non-standard elastic problems. Besides, they present some interest from the purely geometrical point of view.