Hamiltonian dynamics of planar affinely-rigid body

The general formulation of the mechanics of an affinely-rigid body in n dimensions was presented in [1, 2, 3]. Obviously, it is the special case n = 3 that is directly physically applicable, if a proper potential model is chosen. For realistic potentials, the three-dimensional problem is very difficult. The reason is that the group SO(3, IR) (and generally SO(n, IR) for n > 2) is semisimple and because of this the deformative degrees of freedom in kinetic energy form are mixed in a very malicious, non-separable way. Some general aspects of the two-dimensional model were investigated in [4]. The two-dimensional study may be also useful as a preliminary step towards the analysis of realistic three-dimensional problems. Here we consider some isotropic dynamical models in two dimensions. They are both physically reasonable (e.g. from the point of view of macroscopic elasticity) and analytically treatable in terms of the separation of variables method (Stäckel theorem). Some expressions for the action-angle variables are derived, in particular, the dependence of energy on the action parameters is discussed. Our calculations are based on the method of complex integration, elaborated in this context by Max Born [5]. The degeneracy of these models is explicitly described and Bohr-Sommerfeld quantization is performed.