Action-Angle Analysis of Some Geometric Models of Internal Degrees of Freedom

Abstract

We derive and discuss equations of motion of infinitesimal affinely-rigid body moving in Riemannian spaces. There is no concept of extended rigid and affinely rigid body in a general Riemannian space. Therefore the gyroscopes with affine degrees of freedom are described as moving bases attached to the material point. This base is a remnant of extended rigid and affinely rigid body in a flat space. The special stress is laid on affinely rigid bodies in two-dimensional constant curvature spaces (sphere and pseudosphere). In particular, we consider incompressible affinely rigid bodies, like, e.g. fat spots on a water surface (e.g. petrol pollution). This is a two-dimensional analogue of three-dimensional incompressible objects like fluid droplets.