Complex Representation of Planar Motions and Conserved Quantities of the Kepler and Hooke Problems

Abstract

Using a complex representation of planar motions, we show that the dynamical conserved quantities associated to the isotropic harmonic oscillator (Fradkin-Jauch-Hill tensor) and to the Kepler's problem (Laplace-Runge-Lenz vector) find a very simple and natural interpretation. In this frame we also establish in an elementary way the relation which connects them.

Block regularisation of the logarithm central problem

“…Specifically, identifying the plane of the motion with the complex plane, this is achieved with the help of the conformal transformation z → z n . (For more on conformal transformations applied to planar central force problems, see [GBM08].) Our attempts in finding a regularising conformal transformation in the case of the logarithm potential were not successful; this problem remains open.…”

“…For this purpose, the singularity at the origin is blown-up into an invariant collision manifold pasted into the phase space for all levels of energy, and the re-parametrized flow is complete. The regularisation of the Kepler problem and connections with various mathematical-physics fields was, and continues to be, the subject of a multitude of papers: [Mo70,Mil83,HedL12,GBM08,San09,LZ15,vdM21,ChHs22], to name a few.…”

Power-duality in path integral formulation of quantum mechanics

“…There are many ways to convert Newtonʼs law to Hookeʼs law via various transformations of real numbers, complex numbers, spinors, quaternions, hypercomplex numbers and so on. See references [8][9][10][11][12][13][14][15][16][17][18]. These efforts on regularization is indeed all based on recognition of the dual relation between Newtonʼs law and Hookeʼs law.…”

“…According to Chandrasekharʼs reading [9] out of propositions and corollaries in Principia, Newton had even established duality between the centripetal forces of the form r a and r b for various pairs (a, b). The power-law duality for arbitrary power forces in classical mechanics was analyzed by Kasner [13], Arnol'd [8], and others [14][15][16]. In our previous work [1] we expanded the domain of the dual pairs.…”