American Institute of Mathematical Sciences

September  2018, 10(3): 359-372. doi: 10.3934/jgm.2018013

Alternative angle-based approach to the $\mathcal{KS}$-Map. An interpretation through symmetry and reduction

 1 Space Dynamics Group, DITEC, Facultad Informática, Universidad de Murcia, 30100 Campus de Espinardo. Murcia, Spain 2 Grupo GISDA, Dept. de Matemática, Facultad de Ciencias, Universidad del Bío-Bío, Av. Collao 1202. Concepción, Chile

* Corresponding author

Received  October 2017 Revised  May 2018 Published  August 2018

Fund Project: Support came from MTM2015-64095-P, ESP2013-41634-P and FONDECYT 11160224.

The $\mathcal{KS}$ map is revisited in terms of an $S^1$-action in $T^*\mathbb{H}_0$ with the bilinear function as the associated momentum map. Indeed, the $\mathcal{KS}$ transformation maps the $S^1$-fibers related to the mentioned action to single points. By means of this perspective a second twin-bilinear function is obtained with an analogous $S^1$-action. We also show that the connection between the 4-D isotropic harmonic oscillator and the spatial Kepler systems can be done in a straightforward way after regularization and through the extension to 4 degrees of freedom of the Euler angles, when the bilinear relation is imposed. This connection incorporates both bilinear functions among the variables. We will show that an alternative regularization separates the oscillator expressed in Projective Euler variables. This setting takes advantage of the two bilinear functions and another integral of the system including them among a new set of variables that allows to connect the 4-D isotropic harmonic oscillator and the planar Kepler system. In addition, our approach makes transparent that only when we refer to rectilinear solutions, both bilinear relations defining the $\mathcal{KS}$ transformations are needed.

Citation: Sebastián Ferrer, Francisco Crespo. Alternative angle-based approach to the $\mathcal{KS}$-Map. An interpretation through symmetry and reduction. Journal of Geometric Mechanics, 2018, 10 (3) : 359-372. doi: 10.3934/jgm.2018013
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References:
Commutative diagram. The map $\Gamma$ is the transformation from spherical to Cartesian coordinates and $\pi$ is the projection $(\rho, \phi, \theta, \psi, R, \Phi, \Theta, \Psi)\rightarrow (\rho, \phi, \theta, R, \Phi, \Theta)$
Commutative diagram. The map $\Sigma$ is the transformation from polar to Cartesian coordinates and $\pi$ is the projection $(\rho, \lambda, \mu, \nu, R, \Lambda, M, N)\rightarrow (\rho, \mu, R, M)$
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