CENTROIDAL FRAMES IN DYNAMIC-SYSTEMS .1. POINT VORTICES

The dynamics of point vortices is studied in Part I of the paper. It is well known that the (translational) centre-of-mass frame decomposes the motion of a mechanical system into simpler components. It is less known, however, that special rotational frames have also been suggested for the same purpose. In contrast to the centre-of-mass frame, the angular velocities of these rotational frames are not given explicitly that limits their application to small perturbations of rigid body rotations. A new class of centroidal frames (CF) related to different groups such as translation, rotation, dilation, etc., is introduced in this paper. The CFs decompose the motion of point vortices into a group and a relative components without restriction to small perturbations of pure group motions. The definition of the CFs is based on an averaging of motion or on minimization of energy of the relative motion, where an appropriate energy function is expressed through generators of the group action. As a result, the linear and angular velocities as well as other characteristics of the CFs can be obtained explicitly. Part I of the paper presents application of the CFs to a hamiltonian system of point vortices. Examples of integrable and chaotic motions in the CFs visualize dynamical patterns that are completely hidden in the conventional fixed frame (FF). Motions which look like chaotic in the FF reveal a variety of stable and unstable structures in the CFs. Quasi-periodic and chaotic motions coexist for all energies and the CFs permit to clearly distinguish between them. A new phenomenon of asymptotic symmetries (in rotational CFs) Of some chaotic motions is discovered. This is related to a permutation symmetry of the hamiltonian.