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
References:
[1]

A. BarutC. Schneider and R. Wilson, Quantum theory of infinite quantum theory of infinite component fields, J. Math. Phys., 20 (1979), 2244-2256.  doi: 10.1063/1.524005.  Google Scholar

[2]

S. Breiter and K. Langner, Kustaanheimo-Stiefel transformation with an arbitrary defining vector, Celest. Mech. Dynamical Astron., 128 (2017), 323-342.  doi: 10.1007/s10569-017-9754-z.  Google Scholar

[3]

F. H. J. Cornish, The hydrogen atom and the four-dimensional harmonic oscillator, Journal of Physics A: Mathematical and General, 17 (1984), 323–327. doi: 10.1088/0305-4470/17/2/018.  Google Scholar

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F. Crespo, Hopf Fibration Reduction of a Quartic Model. An Application to Rotational and Orbital Dynamics, PhD. Universidad de Murcia. Google Scholar

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R. Cushman and L. Bates, Global Aspects of Classical Integrable Systems, 2nd edition, Birkhäuser Verlag, Basel, 2015. doi: 10.1007/978-3-0348-0918-4.  Google Scholar

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A. DepritA. Elipe and S. Ferrer, Linearization: Laplace vs. Stiefel, Celestial Mechanics and Dynamical Astronomy, 58 (1994), 151-201.  doi: 10.1007/BF00695790.  Google Scholar

[7]

L. Euler, De motu rectilineo trium corporum se mutuo attrahentium, Novi Commentarii academiae scientiarum Petropolitanae, 11 (1767), 144-151.   Google Scholar

[8]

S. Ferrer, The projective Andoyer transformation and the connection between the 4-d isotropic oscillator and Kepler systems, arXiv: 1011.3000v1 [nlin. SI]. Google Scholar

[9]

S. Ferrer and F. Crespo, Parametric quartic Hamiltonian model. A unified treatment of classic integrable systems, Journal of Geometric Mechanics, 6 (2014), 479-502.  doi: 10.3934/jgm.2014.6.479.  Google Scholar

[10]

H. Goldstein, C. Poole and J. Safko, Classical Mechanics, Addison Wesley, New York, Third edition, 2002. Google Scholar

[11]

W. Heard, Rigid Body Mechanics, WILEY-VCH Verlag GmbH & Co. KGaA, Mathematics, Physics and Applications, 2006. doi: 10.1002/9783527618811.  Google Scholar

[12]

M. Ikeda and Y. Miyachi, On the mathematical structure of the symmetry of some simple dynamical systems, Mathematica Japonica, 15 (1970), 127-142.   Google Scholar

[13]

M. Kibler and P. Winternitz, Dynamical invariance algebra of the hartmann potential, Journal of Physics A: Mathematical and General, 20 (1987), 4097-4108.  doi: 10.1088/0305-4470/20/13/018.  Google Scholar

[14]

J. Kuipers, Quaternions and Rotation Sequences, Princeton University text, Princeton, New Jersey, 1999.  Google Scholar

[15]

M. Kummer, On the regularization of the Kepler problem, Communications in Mathematical Physics, 84 (1982), 133-152.  doi: 10.1007/BF01208375.  Google Scholar

[16]

P. Kustaanheimo, Spinor regularization of the Kepler motion, Annales Universitatis Turkuensis, 73 (1964), 7pp.  Google Scholar

[17]

P. Kustaanheimo and E. Stiefel, Perturbation theory of Kepler motion based on spinor regularization, J. Reine Angew. Math., 218 (1965), 204-219.  doi: 10.1515/crll.1965.218.204.  Google Scholar

[18]

T. Levi-Civita, Sur la régularisation du probléme des trois corps, Acta Mathematica, 42 (1920), 99-144.  doi: 10.1007/BF02404404.  Google Scholar

[19]

T. Ligon and M. Schaaf, On the global symmetry of the classical Kepler problem, Reports on Mathematical Physics, 9 (1976), 281-300.  doi: 10.1016/0034-4877(76)90061-6.  Google Scholar

[20]

J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry, 2nd edition, Springer-Verlag New York, Inc., 1999. doi: 10.1007/978-0-387-21792-5.  Google Scholar

[21]

K. Meyer, G. R. Hall and D. Offin, Introduction to Hamiltonian Dynamical Systems and the N-Body Problem, 2nd Ed, vol. 90 of APM, Applied Mathematical Sciences, Springer, New York, 2009.  Google Scholar

[22]

J. Moser, Regularization of Kepler's problem and the averaging method on a manifold, Communications on Pure and Applied Mathematics, 23 (1970), 609-636.  doi: 10.1002/cpa.3160230406.  Google Scholar

[23]

J. RoaH. Urrutxua and J. Peláez, Stability and chaos in kustaanheimo-stiefel space induced by the hopf fibration, Monthly Notices of the Royal Astronomical Society, 459 (2016), 2444-2454.  doi: 10.1093/mnras/stw780.  Google Scholar

[24]

P. Saha, Interpreting the Kustaanheimo-Stiefel transform in gravitational dynamics, Mon. Not. R. Astron. Soc., 400 (2009), 228-231.  doi: 10.1111/j.1365-2966.2009.15437.x.  Google Scholar

[25]

J. Souriau, Sur la variete de Kepler, Convegno di Geometria Simplettica e Fisica Matematica, 14. Google Scholar

[26]

E. Stiefel and G. Scheifele, Linear and Regular Celestial Mechanics, Springer, Berlin. Google Scholar

[27]

J. van der Meer, The Kepler system as a reduced 4d harmonic oscillator, Journal of Geometry and Physics, 92 (2015), 181-193.  doi: 10.1016/j.geomphys.2015.02.016.  Google Scholar

[28]

M. D. Vivarelli, The KS-transformation in hypercomplex form and the quantization of the negative-energy orbit manifold of the Kepler problem, Celestial mechanics, 36 (1985), 349-364.  doi: 10.1007/BF01227489.  Google Scholar

[29]

J. Waldvogel, Quaternions for regularizing celestial mechanics: The right way, Celest. Mech. Dynamical Astron., 102 (2008), 149-162.  doi: 10.1007/s10569-008-9124-y.  Google Scholar

[30]

L. Zhao, Kustaanheimo-Stiefel regularization and the quadrupolar conjugacy, Regul. Chaot. Dyn., 20 (2015), 19-36.  doi: 10.1134/S1560354715010025.  Google Scholar

show all references

References:
[1]

A. BarutC. Schneider and R. Wilson, Quantum theory of infinite quantum theory of infinite component fields, J. Math. Phys., 20 (1979), 2244-2256.  doi: 10.1063/1.524005.  Google Scholar

[2]

S. Breiter and K. Langner, Kustaanheimo-Stiefel transformation with an arbitrary defining vector, Celest. Mech. Dynamical Astron., 128 (2017), 323-342.  doi: 10.1007/s10569-017-9754-z.  Google Scholar

[3]

F. H. J. Cornish, The hydrogen atom and the four-dimensional harmonic oscillator, Journal of Physics A: Mathematical and General, 17 (1984), 323–327. doi: 10.1088/0305-4470/17/2/018.  Google Scholar

[4]

F. Crespo, Hopf Fibration Reduction of a Quartic Model. An Application to Rotational and Orbital Dynamics, PhD. Universidad de Murcia. Google Scholar

[5]

R. Cushman and L. Bates, Global Aspects of Classical Integrable Systems, 2nd edition, Birkhäuser Verlag, Basel, 2015. doi: 10.1007/978-3-0348-0918-4.  Google Scholar

[6]

A. DepritA. Elipe and S. Ferrer, Linearization: Laplace vs. Stiefel, Celestial Mechanics and Dynamical Astronomy, 58 (1994), 151-201.  doi: 10.1007/BF00695790.  Google Scholar

[7]

L. Euler, De motu rectilineo trium corporum se mutuo attrahentium, Novi Commentarii academiae scientiarum Petropolitanae, 11 (1767), 144-151.   Google Scholar

[8]

S. Ferrer, The projective Andoyer transformation and the connection between the 4-d isotropic oscillator and Kepler systems, arXiv: 1011.3000v1 [nlin. SI]. Google Scholar

[9]

S. Ferrer and F. Crespo, Parametric quartic Hamiltonian model. A unified treatment of classic integrable systems, Journal of Geometric Mechanics, 6 (2014), 479-502.  doi: 10.3934/jgm.2014.6.479.  Google Scholar

[10]

H. Goldstein, C. Poole and J. Safko, Classical Mechanics, Addison Wesley, New York, Third edition, 2002. Google Scholar

[11]

W. Heard, Rigid Body Mechanics, WILEY-VCH Verlag GmbH & Co. KGaA, Mathematics, Physics and Applications, 2006. doi: 10.1002/9783527618811.  Google Scholar

[12]

M. Ikeda and Y. Miyachi, On the mathematical structure of the symmetry of some simple dynamical systems, Mathematica Japonica, 15 (1970), 127-142.   Google Scholar

[13]

M. Kibler and P. Winternitz, Dynamical invariance algebra of the hartmann potential, Journal of Physics A: Mathematical and General, 20 (1987), 4097-4108.  doi: 10.1088/0305-4470/20/13/018.  Google Scholar

[14]

J. Kuipers, Quaternions and Rotation Sequences, Princeton University text, Princeton, New Jersey, 1999.  Google Scholar

[15]

M. Kummer, On the regularization of the Kepler problem, Communications in Mathematical Physics, 84 (1982), 133-152.  doi: 10.1007/BF01208375.  Google Scholar

[16]

P. Kustaanheimo, Spinor regularization of the Kepler motion, Annales Universitatis Turkuensis, 73 (1964), 7pp.  Google Scholar

[17]

P. Kustaanheimo and E. Stiefel, Perturbation theory of Kepler motion based on spinor regularization, J. Reine Angew. Math., 218 (1965), 204-219.  doi: 10.1515/crll.1965.218.204.  Google Scholar

[18]

T. Levi-Civita, Sur la régularisation du probléme des trois corps, Acta Mathematica, 42 (1920), 99-144.  doi: 10.1007/BF02404404.  Google Scholar

[19]

T. Ligon and M. Schaaf, On the global symmetry of the classical Kepler problem, Reports on Mathematical Physics, 9 (1976), 281-300.  doi: 10.1016/0034-4877(76)90061-6.  Google Scholar

[20]

J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry, 2nd edition, Springer-Verlag New York, Inc., 1999. doi: 10.1007/978-0-387-21792-5.  Google Scholar

[21]

K. Meyer, G. R. Hall and D. Offin, Introduction to Hamiltonian Dynamical Systems and the N-Body Problem, 2nd Ed, vol. 90 of APM, Applied Mathematical Sciences, Springer, New York, 2009.  Google Scholar

[22]

J. Moser, Regularization of Kepler's problem and the averaging method on a manifold, Communications on Pure and Applied Mathematics, 23 (1970), 609-636.  doi: 10.1002/cpa.3160230406.  Google Scholar

[23]

J. RoaH. Urrutxua and J. Peláez, Stability and chaos in kustaanheimo-stiefel space induced by the hopf fibration, Monthly Notices of the Royal Astronomical Society, 459 (2016), 2444-2454.  doi: 10.1093/mnras/stw780.  Google Scholar

[24]

P. Saha, Interpreting the Kustaanheimo-Stiefel transform in gravitational dynamics, Mon. Not. R. Astron. Soc., 400 (2009), 228-231.  doi: 10.1111/j.1365-2966.2009.15437.x.  Google Scholar

[25]

J. Souriau, Sur la variete de Kepler, Convegno di Geometria Simplettica e Fisica Matematica, 14. Google Scholar

[26]

E. Stiefel and G. Scheifele, Linear and Regular Celestial Mechanics, Springer, Berlin. Google Scholar

[27]

J. van der Meer, The Kepler system as a reduced 4d harmonic oscillator, Journal of Geometry and Physics, 92 (2015), 181-193.  doi: 10.1016/j.geomphys.2015.02.016.  Google Scholar

[28]

M. D. Vivarelli, The KS-transformation in hypercomplex form and the quantization of the negative-energy orbit manifold of the Kepler problem, Celestial mechanics, 36 (1985), 349-364.  doi: 10.1007/BF01227489.  Google Scholar

[29]

J. Waldvogel, Quaternions for regularizing celestial mechanics: The right way, Celest. Mech. Dynamical Astron., 102 (2008), 149-162.  doi: 10.1007/s10569-008-9124-y.  Google Scholar

[30]

L. Zhao, Kustaanheimo-Stiefel regularization and the quadrupolar conjugacy, Regul. Chaot. Dyn., 20 (2015), 19-36.  doi: 10.1134/S1560354715010025.  Google Scholar

Figure 1.  Commutative diagram. The map $\Gamma$ is the transformation from spherical to Cartesian coordinates and $\pi$ is the projection $(\rho, \phi, \theta, \psi, R, \Phi, \Theta, \Psi)\rightarrow (\rho, \phi, \theta, R, \Phi, \Theta)$
Figure 2.  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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