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Families of canonical transformations by Hamilton-Jacobi-Poincaré equation. Application to rotational and orbital motion
1. | Departamento de Matemática Aplicada, Universidad de Murcia, 30100 Espinardo, Spain |
2. | Real Observatorio de la Armada, ES-11 110 San Fernando |
References:
[1] |
M. H. Andoyer, Sur l'anomalie excentrique et l'anomalie vraie comme éléments canoniques du mouvement elliptique d'après MM. T. Levi-Civita et G.-W. Hill,, Bulletin astronomique, 30 (1913), 425. Google Scholar |
[2] |
M. H. Andoyer, "Cours de Mécanique Céleste,", Cours de Mécanique Céleste, 1 (1923). Google Scholar |
[3] |
G. Benettin, The elements of Hamiltonian perturbation theory,, in, (2005), 1.
|
[4] |
D. Boccaletti and G. Pucacco, "Theory of Orbits. 1: Integrable Systems and Non-perturbative Methods,", Springer-Verlag, (2001).
|
[5] |
J. F. Cariñena, X. Gracia, G. Marmo, E. Martínez, M. Muñoz-Lecanda and N. Román-Roy, Geometric Hamilton-Jacobi theory,, Int. J. Geom. Meth. Mod. Phys., 3 (2006), 1417.
doi: 10.1142/S0219887806001764. |
[6] |
D. E. Chang and J. E. Marsden, Geometric derivation of the Delaunay variables and geometric phases,, Celes. Mech. & Dyn. Astron., 86 (2003), 185.
doi: 10.1023/A:1024174702036. |
[7] |
R. H. Cushman and L. M. Bates, "Global Aspects of Classical Integrable Systems,", Birkhäuser Verlag, (1997).
|
[8] |
Ch. E. Delaunay, "Théorie du Mouvement de la Lune,", Mémoires de l'Academie des Sciences de l'Institut Impérial de France, 28 (1867), 9. Google Scholar |
[9] |
A. Deprit, Free rotation of a rigid body studied in the phase space,, American J. Physics, 35 (1967), 424.
doi: 10.1119/1.1974113. |
[10] |
A. Deprit, The elimination of the parallax in satellite theory,, Celes. Mech., 24 (1981), 111.
doi: 10.1007/BF01229192. |
[11] |
A. Deprit, A note concerning the TR-transformation,, Celes. Mech., 23 (1981), 299.
doi: 10.1007/BF01230743. |
[12] |
A. Deprit and A. Elipe, Complete reduction of the Euler-Poinsot problem,, Journal of the Astronautical Sciences, 41 (1993), 603.
|
[13] |
F. Fassò, The Euler-Poinsot top: A non-commutatively integrable system without global action-angle coordinates,, J. Appl. Math. Phys. (ZAMP), 47 (1996), 953.
doi: 10.1007/BF00920045. |
[14] |
F. Fassò, Superintegrable Hamiltonian systems: Geometry and perturbations,, Acta Applicandae Mathematicae, 87 (2005), 93.
doi: 10.1007/s10440-005-1139-8. |
[15] |
S. Ferrer and M. Lara, "Orbital Canonical Transformations. The Isochronal Family,", XII Jornadas Mecánica Celeste, (2009). Google Scholar |
[16] |
S. Ferrer and M. Lara, Integration of the rotation of an earth-like body as a perturbed spherical rotor,, The Astronomical Journal, 139 (2010), 1899.
doi: 10.1088/0004-6256/139/5/1899. |
[17] |
T. Fukushima, Canonical and universal elements of the rotational motion of a triaxial rigid body,, The Astronomical Journal, 136 (2008), 1728.
doi: 10.1088/0004-6256/136/4/1728. |
[18] |
H. Goldstein, C. Poole and J. Safko, "Classical Mechanics,", 3rd edition, (2002). Google Scholar |
[19] |
M. Hénon, L'amas isochrone I,, Annales d'Astrophysique, 22 (1959), 126. Google Scholar |
[20] |
G. W. Hill, Motion of a system of material points under the action of gravitation,, The Astronomical Journal, 27 (1913), 171.
doi: 10.1086/103991. |
[21] |
D. L. Hitzl and J. V. Breakwell, Resonant and non-resonant gravity-gradient perturbations of a tumbling tri-axial satellite,, Celes. Mech., 3 (1971), 346.
doi: 10.1007/BF01231806. |
[22] |
D. Iglesias, M. de León and D. Martín de Diego, Towards a Hamilton-Jacobi theory for nonholonomic mechanical systems,, J. Phys. A: Math. Theor., 41 (2008).
doi: 10.1088/1751-8113/41/1/015205. |
[23] |
J. V. José and E. J. Saletan, "Classical Dynamics: A Contemporary Approach,", Cambridge University Press, (2000).
|
[24] |
H. Kinoshita, First-order perturbations of the two finite body problem,, Publications of the Astronomical Society of Japan, 24 (1972), 423. Google Scholar |
[25] |
V. V. Kozlov, Geometry of the "action-angle" variables in the Euler-Poinsot problem,, Vestnik Moskov. Univ., 29 (1974), 74.
|
[26] |
M. de León, J. C. Marrero and D. Martin de Diego, Linear almost Poisson structures and Hamilton-Jacobi equation. Application to nonholonomic mechanics,, Journal of Geometric Mechanics, 2 (2010), 159.
doi: 10.3934/jgm.2010.2.159. |
[27] |
T. Levi-Civita, Nouvo sistema canonico di elementi ellittici,, Annali di Matematica Serie III, XX (1913), 153.
doi: 10.1007/BF02419588. |
[28] |
T. Levi-Civita, Sur la régularization du problème des trois corps,, Acta Mathematica, 42 (1918), 99.
doi: 10.1007/BF02404404. |
[29] |
R. de la Llave, A. González, A. Jorba and J. Villanueva, KAM theory without action-angle variables,, Nonlinearity, 18 (2005), 855.
doi: 10.1088/0951-7715/18/2/020. |
[30] |
J. E. Marsden and T. Ratiu, "Introduction to Mechanics and Symmetry,", TAM \textbf{17}, 17 (1999).
|
[31] |
K. Meyer, G. R. Hall and D. Offin, "Introduction to Hamiltonian Dynamical Systems and the N-Body Problem,", 2nd edition, 90 (2009).
|
[32] |
J. Moser, Regularization of Kepler's problem and the averaging method on a manifold,, Comm. Pure Appl. Math., 23 (1970), 609.
doi: 10.1002/cpa.3160230406. |
[33] |
J. Moser and E. J. Zehnder, "Notes on Dynamical Systems,", Courant Lectures Notes in Mathematics, 12 (2005).
|
[34] |
J. P. Ortega and T. Ratiu, "Momentum Maps and Hamiltonian Reduction,", Progress in Mathematics, (2004).
|
[35] |
H. Poincaré, "Les Méthodes Mouvelles de la Mécanique Céleste," 2,, Gauthier-Villars et Fils, (1893), 315. Google Scholar |
[36] |
Yu. A. Sadov, "The Action-Angles Variables in the Euler-Poinsot Problem,", Prikladnaya Matematika i Mekhanika, 34 (1970), 962. Google Scholar |
[37] |
Yu. A. Sadov, "The Action-Angle Variables in the Euler-Poinsot Problem,", Preprint No. \textbf{22} of the Keldysh Institute of Applied Mathematics of the Academy of Sciences of the USSR, 22 (1970). Google Scholar |
[38] |
G. Scheifele, Généralisation des éléments de Delaunay en mécanique céleste. Application au mouvement d'un satellite artificiel,, Les Comptes rendus de l'Académie des sciences de Paris Sér. A, 271 (1970), 729.
|
[39] |
J. Struckmeier, Hamiltonian Dynamics on the symplectic extended phase space for autonomous and non-autonomous systems,, Journal of Physics A: Mathematical and General, 38 (2005), 1257.
doi: 10.1088/0305-4470/38/6/006. |
[40] |
G. J. Sussman, J. Wisdom and M. Mayer, "Structure and Interpretation of Classical Mechanics,", The MIT Press, (2001).
|
[41] |
E. T. Whittaker, "A Treatise on the Analytical Dynamics of Particles and Rigid Bodies,", Cambridge Mathematical Library, (1988).
|
[42] |
P. Yanguas, Perturbations of the isochrone model,, Nonlinearity, 14 (2001), 1.
doi: 10.1088/0951-7715/14/1/301. |
show all references
References:
[1] |
M. H. Andoyer, Sur l'anomalie excentrique et l'anomalie vraie comme éléments canoniques du mouvement elliptique d'après MM. T. Levi-Civita et G.-W. Hill,, Bulletin astronomique, 30 (1913), 425. Google Scholar |
[2] |
M. H. Andoyer, "Cours de Mécanique Céleste,", Cours de Mécanique Céleste, 1 (1923). Google Scholar |
[3] |
G. Benettin, The elements of Hamiltonian perturbation theory,, in, (2005), 1.
|
[4] |
D. Boccaletti and G. Pucacco, "Theory of Orbits. 1: Integrable Systems and Non-perturbative Methods,", Springer-Verlag, (2001).
|
[5] |
J. F. Cariñena, X. Gracia, G. Marmo, E. Martínez, M. Muñoz-Lecanda and N. Román-Roy, Geometric Hamilton-Jacobi theory,, Int. J. Geom. Meth. Mod. Phys., 3 (2006), 1417.
doi: 10.1142/S0219887806001764. |
[6] |
D. E. Chang and J. E. Marsden, Geometric derivation of the Delaunay variables and geometric phases,, Celes. Mech. & Dyn. Astron., 86 (2003), 185.
doi: 10.1023/A:1024174702036. |
[7] |
R. H. Cushman and L. M. Bates, "Global Aspects of Classical Integrable Systems,", Birkhäuser Verlag, (1997).
|
[8] |
Ch. E. Delaunay, "Théorie du Mouvement de la Lune,", Mémoires de l'Academie des Sciences de l'Institut Impérial de France, 28 (1867), 9. Google Scholar |
[9] |
A. Deprit, Free rotation of a rigid body studied in the phase space,, American J. Physics, 35 (1967), 424.
doi: 10.1119/1.1974113. |
[10] |
A. Deprit, The elimination of the parallax in satellite theory,, Celes. Mech., 24 (1981), 111.
doi: 10.1007/BF01229192. |
[11] |
A. Deprit, A note concerning the TR-transformation,, Celes. Mech., 23 (1981), 299.
doi: 10.1007/BF01230743. |
[12] |
A. Deprit and A. Elipe, Complete reduction of the Euler-Poinsot problem,, Journal of the Astronautical Sciences, 41 (1993), 603.
|
[13] |
F. Fassò, The Euler-Poinsot top: A non-commutatively integrable system without global action-angle coordinates,, J. Appl. Math. Phys. (ZAMP), 47 (1996), 953.
doi: 10.1007/BF00920045. |
[14] |
F. Fassò, Superintegrable Hamiltonian systems: Geometry and perturbations,, Acta Applicandae Mathematicae, 87 (2005), 93.
doi: 10.1007/s10440-005-1139-8. |
[15] |
S. Ferrer and M. Lara, "Orbital Canonical Transformations. The Isochronal Family,", XII Jornadas Mecánica Celeste, (2009). Google Scholar |
[16] |
S. Ferrer and M. Lara, Integration of the rotation of an earth-like body as a perturbed spherical rotor,, The Astronomical Journal, 139 (2010), 1899.
doi: 10.1088/0004-6256/139/5/1899. |
[17] |
T. Fukushima, Canonical and universal elements of the rotational motion of a triaxial rigid body,, The Astronomical Journal, 136 (2008), 1728.
doi: 10.1088/0004-6256/136/4/1728. |
[18] |
H. Goldstein, C. Poole and J. Safko, "Classical Mechanics,", 3rd edition, (2002). Google Scholar |
[19] |
M. Hénon, L'amas isochrone I,, Annales d'Astrophysique, 22 (1959), 126. Google Scholar |
[20] |
G. W. Hill, Motion of a system of material points under the action of gravitation,, The Astronomical Journal, 27 (1913), 171.
doi: 10.1086/103991. |
[21] |
D. L. Hitzl and J. V. Breakwell, Resonant and non-resonant gravity-gradient perturbations of a tumbling tri-axial satellite,, Celes. Mech., 3 (1971), 346.
doi: 10.1007/BF01231806. |
[22] |
D. Iglesias, M. de León and D. Martín de Diego, Towards a Hamilton-Jacobi theory for nonholonomic mechanical systems,, J. Phys. A: Math. Theor., 41 (2008).
doi: 10.1088/1751-8113/41/1/015205. |
[23] |
J. V. José and E. J. Saletan, "Classical Dynamics: A Contemporary Approach,", Cambridge University Press, (2000).
|
[24] |
H. Kinoshita, First-order perturbations of the two finite body problem,, Publications of the Astronomical Society of Japan, 24 (1972), 423. Google Scholar |
[25] |
V. V. Kozlov, Geometry of the "action-angle" variables in the Euler-Poinsot problem,, Vestnik Moskov. Univ., 29 (1974), 74.
|
[26] |
M. de León, J. C. Marrero and D. Martin de Diego, Linear almost Poisson structures and Hamilton-Jacobi equation. Application to nonholonomic mechanics,, Journal of Geometric Mechanics, 2 (2010), 159.
doi: 10.3934/jgm.2010.2.159. |
[27] |
T. Levi-Civita, Nouvo sistema canonico di elementi ellittici,, Annali di Matematica Serie III, XX (1913), 153.
doi: 10.1007/BF02419588. |
[28] |
T. Levi-Civita, Sur la régularization du problème des trois corps,, Acta Mathematica, 42 (1918), 99.
doi: 10.1007/BF02404404. |
[29] |
R. de la Llave, A. González, A. Jorba and J. Villanueva, KAM theory without action-angle variables,, Nonlinearity, 18 (2005), 855.
doi: 10.1088/0951-7715/18/2/020. |
[30] |
J. E. Marsden and T. Ratiu, "Introduction to Mechanics and Symmetry,", TAM \textbf{17}, 17 (1999).
|
[31] |
K. Meyer, G. R. Hall and D. Offin, "Introduction to Hamiltonian Dynamical Systems and the N-Body Problem,", 2nd edition, 90 (2009).
|
[32] |
J. Moser, Regularization of Kepler's problem and the averaging method on a manifold,, Comm. Pure Appl. Math., 23 (1970), 609.
doi: 10.1002/cpa.3160230406. |
[33] |
J. Moser and E. J. Zehnder, "Notes on Dynamical Systems,", Courant Lectures Notes in Mathematics, 12 (2005).
|
[34] |
J. P. Ortega and T. Ratiu, "Momentum Maps and Hamiltonian Reduction,", Progress in Mathematics, (2004).
|
[35] |
H. Poincaré, "Les Méthodes Mouvelles de la Mécanique Céleste," 2,, Gauthier-Villars et Fils, (1893), 315. Google Scholar |
[36] |
Yu. A. Sadov, "The Action-Angles Variables in the Euler-Poinsot Problem,", Prikladnaya Matematika i Mekhanika, 34 (1970), 962. Google Scholar |
[37] |
Yu. A. Sadov, "The Action-Angle Variables in the Euler-Poinsot Problem,", Preprint No. \textbf{22} of the Keldysh Institute of Applied Mathematics of the Academy of Sciences of the USSR, 22 (1970). Google Scholar |
[38] |
G. Scheifele, Généralisation des éléments de Delaunay en mécanique céleste. Application au mouvement d'un satellite artificiel,, Les Comptes rendus de l'Académie des sciences de Paris Sér. A, 271 (1970), 729.
|
[39] |
J. Struckmeier, Hamiltonian Dynamics on the symplectic extended phase space for autonomous and non-autonomous systems,, Journal of Physics A: Mathematical and General, 38 (2005), 1257.
doi: 10.1088/0305-4470/38/6/006. |
[40] |
G. J. Sussman, J. Wisdom and M. Mayer, "Structure and Interpretation of Classical Mechanics,", The MIT Press, (2001).
|
[41] |
E. T. Whittaker, "A Treatise on the Analytical Dynamics of Particles and Rigid Bodies,", Cambridge Mathematical Library, (1988).
|
[42] |
P. Yanguas, Perturbations of the isochrone model,, Nonlinearity, 14 (2001), 1.
doi: 10.1088/0951-7715/14/1/301. |
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