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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), 425-429. |
[2] |
M. H. Andoyer, "Cours de Mécanique Céleste," 1, Gauthier-Villars et cie, Paris, 1923. (p 57) |
[3] |
G. Benettin, The elements of Hamiltonian perturbation theory, in "Hamiltonian Systems and Fourier Analysis: New Prospects for Gravitational Dynamics" (eds. D. Benest and C. Froeschle), Cambridge Scientific Publ., (2005), 1-98. |
[4] |
D. Boccaletti and G. Pucacco, "Theory of Orbits. 1: Integrable Systems and Non-perturbative Methods," Springer-Verlag, Berlin, 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), 1417-1458.
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), 185-208.
doi: 10.1023/A:1024174702036. |
[7] |
R. H. Cushman and L. M. Bates, "Global Aspects of Classical Integrable Systems," Birkhäuser Verlag, Basel, 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. (see Chap. 1, para. 5, pp. 9-11) |
[9] |
A. Deprit, Free rotation of a rigid body studied in the phase space, American J. Physics, 35 (1967), 424-428.
doi: 10.1119/1.1974113. |
[10] |
A. Deprit, The elimination of the parallax in satellite theory, Celes. Mech., 24 (1981), 111-153.
doi: 10.1007/BF01229192. |
[11] |
A. Deprit, A note concerning the TR-transformation, Celes. Mech., 23 (1981), 299-305.
doi: 10.1007/BF01230743. |
[12] |
A. Deprit and A. Elipe, Complete reduction of the Euler-Poinsot problem, Journal of the Astronautical Sciences, 41 (1993), 603-628. |
[13] |
F. Fassò, The Euler-Poinsot top: A non-commutatively integrable system without global action-angle coordinates, J. Appl. Math. Phys. (ZAMP), 47 (1996), 953-976.
doi: 10.1007/BF00920045. |
[14] |
F. Fassò, Superintegrable Hamiltonian systems: Geometry and perturbations, Acta Applicandae Mathematicae, 87 (2005), 93-121.
doi: 10.1007/s10440-005-1139-8. |
[15] |
S. Ferrer and M. Lara, "Orbital Canonical Transformations. The Isochronal Family," XII Jornadas Mecánica Celeste, Lalín, Spain, 2009. |
[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), 1899-1908.
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), 1728-1735.
doi: 10.1088/0004-6256/136/4/1728. |
[18] |
H. Goldstein, C. Poole and J. Safko, "Classical Mechanics," 3rd edition, Pearson Addison-Wesley, 2002. |
[19] |
M. Hénon, L'amas isochrone I, Annales d'Astrophysique, 22 (1959), 126-139. |
[20] |
G. W. Hill, Motion of a system of material points under the action of gravitation, The Astronomical Journal, 27 (1913), 171-182.
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), 346-383.
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), 015205.
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), 423-457. |
[25] |
V. V. Kozlov, Geometry of the "action-angle" variables in the Euler-Poinsot problem, Vestnik Moskov. Univ., Ser. I Mat. Mekh., 29 (1974), no. 5, 74-79. (in Russian) |
[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), 159-198.
doi: 10.3934/jgm.2010.2.159. |
[27] |
T. Levi-Civita, Nouvo sistema canonico di elementi ellittici, Annali di Matematica Serie III, XX (1913), 153-170.
doi: 10.1007/BF02419588. |
[28] |
T. Levi-Civita, Sur la régularization du problème des trois corps, Acta Mathematica, 42 (1918), 99-144.
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), 855-895.
doi: 10.1088/0951-7715/18/2/020. |
[30] |
J. E. Marsden and T. Ratiu, "Introduction to Mechanics and Symmetry," TAM 17, 2nd edition, Springer, New York, 1999. |
[31] |
K. Meyer, G. R. Hall and D. Offin, "Introduction to Hamiltonian Dynamical Systems and the N-Body Problem," 2nd edition, Applied Mathematical Sciences, 90, Springer, New York, 2009. |
[32] |
J. Moser, Regularization of Kepler's problem and the averaging method on a manifold, Comm. Pure Appl. Math., 23 (1970), 609-636.
doi: 10.1002/cpa.3160230406. |
[33] |
J. Moser and E. J. Zehnder, "Notes on Dynamical Systems," Courant Lectures Notes in Mathematics, 12, AMS, Providence, 2005. |
[34] |
J. P. Ortega and T. Ratiu, "Momentum Maps and Hamiltonian Reduction," Progress in Mathematics, 222. Birkhäuser, Boston, 2004. |
[35] |
H. Poincaré, "Les Méthodes Mouvelles de la Mécanique Céleste," 2, Gauthier-Villars et Fils, Paris, 1893, pp. 315-342. |
[36] |
Yu. A. Sadov, "The Action-Angles Variables in the Euler-Poinsot Problem," Prikladnaya Matematika i Mekhanika, 34 (1970), 962-964. [Journal of Applied Mathematics and Mechanics (English translation), 34 (1970), 922-925.] |
[37] |
Yu. A. Sadov, "The Action-Angle Variables in the Euler-Poinsot Problem," Preprint No. 22 of the Keldysh Institute of Applied Mathematics of the Academy of Sciences of the USSR, Moscow (in Russian), 1970. |
[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), 729-732. |
[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), 1257-1278.
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, Cambridge, 2001. |
[41] |
E. T. Whittaker, "A Treatise on the Analytical Dynamics of Particles and Rigid Bodies," Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1988. (reprint of the 1937 edition) |
[42] |
P. Yanguas, Perturbations of the isochrone model, Nonlinearity, 14 (2001), 1-34.
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), 425-429. |
[2] |
M. H. Andoyer, "Cours de Mécanique Céleste," 1, Gauthier-Villars et cie, Paris, 1923. (p 57) |
[3] |
G. Benettin, The elements of Hamiltonian perturbation theory, in "Hamiltonian Systems and Fourier Analysis: New Prospects for Gravitational Dynamics" (eds. D. Benest and C. Froeschle), Cambridge Scientific Publ., (2005), 1-98. |
[4] |
D. Boccaletti and G. Pucacco, "Theory of Orbits. 1: Integrable Systems and Non-perturbative Methods," Springer-Verlag, Berlin, 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), 1417-1458.
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), 185-208.
doi: 10.1023/A:1024174702036. |
[7] |
R. H. Cushman and L. M. Bates, "Global Aspects of Classical Integrable Systems," Birkhäuser Verlag, Basel, 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. (see Chap. 1, para. 5, pp. 9-11) |
[9] |
A. Deprit, Free rotation of a rigid body studied in the phase space, American J. Physics, 35 (1967), 424-428.
doi: 10.1119/1.1974113. |
[10] |
A. Deprit, The elimination of the parallax in satellite theory, Celes. Mech., 24 (1981), 111-153.
doi: 10.1007/BF01229192. |
[11] |
A. Deprit, A note concerning the TR-transformation, Celes. Mech., 23 (1981), 299-305.
doi: 10.1007/BF01230743. |
[12] |
A. Deprit and A. Elipe, Complete reduction of the Euler-Poinsot problem, Journal of the Astronautical Sciences, 41 (1993), 603-628. |
[13] |
F. Fassò, The Euler-Poinsot top: A non-commutatively integrable system without global action-angle coordinates, J. Appl. Math. Phys. (ZAMP), 47 (1996), 953-976.
doi: 10.1007/BF00920045. |
[14] |
F. Fassò, Superintegrable Hamiltonian systems: Geometry and perturbations, Acta Applicandae Mathematicae, 87 (2005), 93-121.
doi: 10.1007/s10440-005-1139-8. |
[15] |
S. Ferrer and M. Lara, "Orbital Canonical Transformations. The Isochronal Family," XII Jornadas Mecánica Celeste, Lalín, Spain, 2009. |
[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), 1899-1908.
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), 1728-1735.
doi: 10.1088/0004-6256/136/4/1728. |
[18] |
H. Goldstein, C. Poole and J. Safko, "Classical Mechanics," 3rd edition, Pearson Addison-Wesley, 2002. |
[19] |
M. Hénon, L'amas isochrone I, Annales d'Astrophysique, 22 (1959), 126-139. |
[20] |
G. W. Hill, Motion of a system of material points under the action of gravitation, The Astronomical Journal, 27 (1913), 171-182.
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), 346-383.
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), 015205.
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), 423-457. |
[25] |
V. V. Kozlov, Geometry of the "action-angle" variables in the Euler-Poinsot problem, Vestnik Moskov. Univ., Ser. I Mat. Mekh., 29 (1974), no. 5, 74-79. (in Russian) |
[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), 159-198.
doi: 10.3934/jgm.2010.2.159. |
[27] |
T. Levi-Civita, Nouvo sistema canonico di elementi ellittici, Annali di Matematica Serie III, XX (1913), 153-170.
doi: 10.1007/BF02419588. |
[28] |
T. Levi-Civita, Sur la régularization du problème des trois corps, Acta Mathematica, 42 (1918), 99-144.
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), 855-895.
doi: 10.1088/0951-7715/18/2/020. |
[30] |
J. E. Marsden and T. Ratiu, "Introduction to Mechanics and Symmetry," TAM 17, 2nd edition, Springer, New York, 1999. |
[31] |
K. Meyer, G. R. Hall and D. Offin, "Introduction to Hamiltonian Dynamical Systems and the N-Body Problem," 2nd edition, Applied Mathematical Sciences, 90, Springer, New York, 2009. |
[32] |
J. Moser, Regularization of Kepler's problem and the averaging method on a manifold, Comm. Pure Appl. Math., 23 (1970), 609-636.
doi: 10.1002/cpa.3160230406. |
[33] |
J. Moser and E. J. Zehnder, "Notes on Dynamical Systems," Courant Lectures Notes in Mathematics, 12, AMS, Providence, 2005. |
[34] |
J. P. Ortega and T. Ratiu, "Momentum Maps and Hamiltonian Reduction," Progress in Mathematics, 222. Birkhäuser, Boston, 2004. |
[35] |
H. Poincaré, "Les Méthodes Mouvelles de la Mécanique Céleste," 2, Gauthier-Villars et Fils, Paris, 1893, pp. 315-342. |
[36] |
Yu. A. Sadov, "The Action-Angles Variables in the Euler-Poinsot Problem," Prikladnaya Matematika i Mekhanika, 34 (1970), 962-964. [Journal of Applied Mathematics and Mechanics (English translation), 34 (1970), 922-925.] |
[37] |
Yu. A. Sadov, "The Action-Angle Variables in the Euler-Poinsot Problem," Preprint No. 22 of the Keldysh Institute of Applied Mathematics of the Academy of Sciences of the USSR, Moscow (in Russian), 1970. |
[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), 729-732. |
[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), 1257-1278.
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, Cambridge, 2001. |
[41] |
E. T. Whittaker, "A Treatise on the Analytical Dynamics of Particles and Rigid Bodies," Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1988. (reprint of the 1937 edition) |
[42] |
P. Yanguas, Perturbations of the isochrone model, Nonlinearity, 14 (2001), 1-34.
doi: 10.1088/0951-7715/14/1/301. |
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