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Dirac pairs
A property of conformally Hamiltonian vector fields; Application to the Kepler problem
1. | Université Pierre et Marie Curie, Institut de mathématiques de Jussieu, 4 place Jussieu, case courrier 247, 75252 Paris cedex 05, France |
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. |
show all references
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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