On the connection among three classical mechanical problems via the hypercomplex KS-transformation

Abstract

With reference to our previous notes [23–30] on the hypercomplex regularizations of the classical Kepler problem and on the hypercomplex descriptions of the classical rotational mechanics, we arrive at an intimate mathematical relationship among the three classical mechanical problems: the elliptic Kepler problem, the spherical Euler-Poinsot problem and the four-dimensional isotropic harmonic oscillator. A key role is played: (i) by the Kustaanheimo-Stiefel (KS) regularizing transformation viewed, in the light of Souriau's quantization procedure, as the projection map in the Hopf fibering of the contact 3-sphere; (ii) by a novel Lagrangian description of rotational kinematics expressed in terms of the Euler-Rodrigues parameters; (iii) by the unit vector a (characterizing both the attitude frame of the rotator and the direction of the major axis of the Kepler orbit) and shown to link ‘a la Cartan’ the KS-theory with the rotation theory. The Kepler-Poinsot-Oscillator connection is examined also in a quantistic Schroedinger approach and is related to the results of Ikeda and Miyachi.