A property of conformally Hamiltonian vector fields; Application to the Kepler problem

June 2012, 4(2): 181-206. doi: 10.3934/jgm.2012.4.181

A property of conformally Hamiltonian vector fields; Application to the Kepler problem

Charles-Michel Marle ^1,

1.	Université Pierre et Marie Curie, Institut de mathématiques de Jussieu, 4 place Jussieu, case courrier 247, 75252 Paris cedex 05, France

Received November 2010 Revised February 2011 Published August 2012

Abstract
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Let $X$ be a Hamiltonian vector field defined on a symplectic manifold $(M,\omega)$, $g$ a nowhere vanishing smooth function defined on an open dense subset $M^0$ of $M$. We will say that the vector field $Y=gX$ is \emph{conformally Hamiltonian}. We prove that when $X$ is complete, when $Y$ is Hamiltonian with respect to another symplectic form $\omega_2$ defined on $M^0$, and when another technical condition is satisfied, then there is a symplectic diffeomorphism from $(M^0,\omega_2)$ onto an open subset of $(M,\omega)$, which maps each orbit to itself and is equivariant with respect to the flows of the vector fields $Y$ on $M^0$ and $X$ on $M$. This result explains why the diffeomorphism of the phase space of the Kepler problem restricted to the negative (resp. positive) values of the energy function, onto an open subset of the cotangent bundle to a three-dimensional sphere (resp. two-sheeted hyperboloid), discovered by Györgyi (1968) [10], re-discovered by Ligon and Schaaf (1976) [16], is a symplectic diffeomorphism. Cushman and Duistermaat (1997) [5] have shown that the Györgyi-Ligon-Schaaf diffeomorphism is characterized by three very natural properties; here that diffeomorphism is obtained by composition of the diffeomorphism given by our result about conformally Hamiltonian vector fields with a (non-symplectic) diffeomorphism built by a variant of Moser's method [20]. Infinitesimal symmetries of the Kepler problem are discussed, and it is shown that their space is a Lie algebroid with zero anchor map rather than a Lie algebra.

Keywords: Conformally Hamiltonian vector fields, Kepler problem..

Mathematics Subject Classification: Primary: 70H05; Secondary: 37J10, 53D05, 70H3.

Citation: Charles-Michel Marle. A property of conformally Hamiltonian vector fields; Application to the Kepler problem. Journal of Geometric Mechanics, 2012, 4 (2) : 181-206. doi: 10.3934/jgm.2012.4.181

References:

[1]	D. V. Anosov, A note on the Kepler problem, Journal of Dynamical and Control Systems, 8 (2002), 413-442. doi: 10.1023/A:1016386605889.
[2]	J. Bernoulli, Extrait de la réponse de M. Bernoulli à M. Herman, datée de Basle le 7 octobre 1710, Histoire de l'Académie Royale des Sciences, année M.DCC.X, avec les Memoires de Mathématiques et de Physique pour la même année, (1710), 521-533. Available from: http://gallica.bnf.fr/ark:/12148/bpt6k34901/f707.image.pagination.
[3]	A. Cannas da Silva and A. Weinstein, "Geometric Models for Noncommutative Algebras," Berkeley Mathematics Lecture Notes, 10, Amer. Math. Soc., Providence, RI; Berkeley Center for Pure and Applied Mathematics, Berkeley, CA, 1999.
[4]	R. H. Cushman and L. Bates, "Global Aspects of Classical Integrable Systems," Birkhäuser Verlag, Basel, 1997.
[5]	R. H. Cushman and J. J. Duistermaat, A characterization of the Ligon-Schaaf regularization map, Comm. on Pure and Appl. Math., 50 (1997), 773-787.
[6]	A. Douady and M. Lazard, Espaces fibrés en algèbres de Lie et en groupes, (French) [Fibered spaces in Lie algebras and in groups], Invent. Math., 1 (1966), 133-151.
[7]	V. A. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik, 98 (1935), 145-154. doi: 10.1007/BF01336904.
[8]	D. Goodstein, J. Goodstein and R. Feynman, "Feynman's Lost Lecture. The Motion of Planets Around the Sun," W. W. Norton and Company Inc., New York, 1996; French translation: Cassini, Paris, 2009.
[9]	A. Guichardet, "Le problème de Kepler; histoire et théorie," Éditions de l'École Polytechnique, Paris, 2012.
[10]	G. Györgyi, Kepler's equation, Fock variables, Bacry's generators and Dirac brackets, parts I and II, Il Nuovo Cimento, 53 (1968), 717-736, and 62 (1969), 449-474.
[11]	W. R. Hamilton, The hodograph or a new method of expressing in symbolic language the Newtonian law of attraction, Proc. Roy. Irish Acad., 3 (1846), 287-294.
[12]	G. Heckman and T. de Laat, On the regularization of the Kepler problem,, preprint, ().
[13]	J. Herman, Extrait d'une lettre de M. Herman à M. Bernoulli, datée de Padoüe le 12 juillet 1710, Histoire de l'Académie Royale des Sciences, année M.DCC.X, avec les Memoires de Mathématiques et de Physique pour la même année, (1710), 519-521. Available from: http://gallica.bnf.fr/ark:/12148/bpt6k34901/f709.image.pagination.
[14]	T. Levi-Civita, Sur la résolution qualitative du problème restreint des trois corps, Acta Math., 30 (1906), 305-327. doi: 10.1007/BF02418577.
[15]	P. Libermann and C.-M. Marle, "Symplectic Geometry and Analytical Mechanics," Mathematics and its Applications, 35, D. Reidel Publishing Company, Dordrecht, 1987. doi: 10.1007/978-94-009-3807-6.
[16]	T. Ligon and M. Schaaf, On the global symmetry of the classical Kepler problem, Rep. Math. Phys., 9 (1976), 281-300. doi: 10.1016/0034-4877(76)90061-6.
[17]	A. J. Maciejewski, M. Prybylska and A. V. Tsiganov, On algebraic construction of certain integrable and super-integrable systems,, preprint, ().
[18]	K. C. H. Mackenzie, "General Theory of Lie Groupoids and Lie Algebroids," London Mathematical Society lecture Note Series, 213, Cambridge University Press, Cambridge, 2005.
[19]	J. Milnor, On the geometry of the Kepler problem, Amer. Math. Monthly, 90 (1983), 353-365. doi: 10.2307/2975570.
[20]	J. Moser, Regularization of Kepler's problem and the averaging method on a manifold, Commun. Pure Appl. Math., 23 (1970), 609-636. doi: 10.1002/cpa.3160230406.
[21]	J.-P. Ortega and T. S. Ratiu, "Momentum Maps and Hamiltonian Reduction," Progress in Mathematics, 222, Birkhäuser Boston, Inc., Boston, MA, 2004.
[22]	J.-M. Souriau, Géométrie globale du problème à deux corps, (French) [Global geometry of the two-body problem], Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur., 117 (1983), 369-418.

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References:

[1]	D. V. Anosov, A note on the Kepler problem, Journal of Dynamical and Control Systems, 8 (2002), 413-442. doi: 10.1023/A:1016386605889.
[2]	J. Bernoulli, Extrait de la réponse de M. Bernoulli à M. Herman, datée de Basle le 7 octobre 1710, Histoire de l'Académie Royale des Sciences, année M.DCC.X, avec les Memoires de Mathématiques et de Physique pour la même année, (1710), 521-533. Available from: http://gallica.bnf.fr/ark:/12148/bpt6k34901/f707.image.pagination.
[3]	A. Cannas da Silva and A. Weinstein, "Geometric Models for Noncommutative Algebras," Berkeley Mathematics Lecture Notes, 10, Amer. Math. Soc., Providence, RI; Berkeley Center for Pure and Applied Mathematics, Berkeley, CA, 1999.
[4]	R. H. Cushman and L. Bates, "Global Aspects of Classical Integrable Systems," Birkhäuser Verlag, Basel, 1997.
[5]	R. H. Cushman and J. J. Duistermaat, A characterization of the Ligon-Schaaf regularization map, Comm. on Pure and Appl. Math., 50 (1997), 773-787.
[6]	A. Douady and M. Lazard, Espaces fibrés en algèbres de Lie et en groupes, (French) [Fibered spaces in Lie algebras and in groups], Invent. Math., 1 (1966), 133-151.
[7]	V. A. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik, 98 (1935), 145-154. doi: 10.1007/BF01336904.
[8]	D. Goodstein, J. Goodstein and R. Feynman, "Feynman's Lost Lecture. The Motion of Planets Around the Sun," W. W. Norton and Company Inc., New York, 1996; French translation: Cassini, Paris, 2009.
[9]	A. Guichardet, "Le problème de Kepler; histoire et théorie," Éditions de l'École Polytechnique, Paris, 2012.
[10]	G. Györgyi, Kepler's equation, Fock variables, Bacry's generators and Dirac brackets, parts I and II, Il Nuovo Cimento, 53 (1968), 717-736, and 62 (1969), 449-474.
[11]	W. R. Hamilton, The hodograph or a new method of expressing in symbolic language the Newtonian law of attraction, Proc. Roy. Irish Acad., 3 (1846), 287-294.
[12]	G. Heckman and T. de Laat, On the regularization of the Kepler problem,, preprint, ().
[13]	J. Herman, Extrait d'une lettre de M. Herman à M. Bernoulli, datée de Padoüe le 12 juillet 1710, Histoire de l'Académie Royale des Sciences, année M.DCC.X, avec les Memoires de Mathématiques et de Physique pour la même année, (1710), 519-521. Available from: http://gallica.bnf.fr/ark:/12148/bpt6k34901/f709.image.pagination.
[14]	T. Levi-Civita, Sur la résolution qualitative du problème restreint des trois corps, Acta Math., 30 (1906), 305-327. doi: 10.1007/BF02418577.
[15]	P. Libermann and C.-M. Marle, "Symplectic Geometry and Analytical Mechanics," Mathematics and its Applications, 35, D. Reidel Publishing Company, Dordrecht, 1987. doi: 10.1007/978-94-009-3807-6.
[16]	T. Ligon and M. Schaaf, On the global symmetry of the classical Kepler problem, Rep. Math. Phys., 9 (1976), 281-300. doi: 10.1016/0034-4877(76)90061-6.
[17]	A. J. Maciejewski, M. Prybylska and A. V. Tsiganov, On algebraic construction of certain integrable and super-integrable systems,, preprint, ().
[18]	K. C. H. Mackenzie, "General Theory of Lie Groupoids and Lie Algebroids," London Mathematical Society lecture Note Series, 213, Cambridge University Press, Cambridge, 2005.
[19]	J. Milnor, On the geometry of the Kepler problem, Amer. Math. Monthly, 90 (1983), 353-365. doi: 10.2307/2975570.
[20]	J. Moser, Regularization of Kepler's problem and the averaging method on a manifold, Commun. Pure Appl. Math., 23 (1970), 609-636. doi: 10.1002/cpa.3160230406.
[21]	J.-P. Ortega and T. S. Ratiu, "Momentum Maps and Hamiltonian Reduction," Progress in Mathematics, 222, Birkhäuser Boston, Inc., Boston, MA, 2004.
[22]	J.-M. Souriau, Géométrie globale du problème à deux corps, (French) [Global geometry of the two-body problem], Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur., 117 (1983), 369-418.

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