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September  2010, 2(3): 223-241. doi: 10.3934/jgm.2010.2.223

Families of canonical transformations by Hamilton-Jacobi-Poincaré equation. Application to rotational and orbital motion

Sebastián Ferrer 1, and  Martin Lara 2,

1. 

Departamento de Matemática Aplicada, Universidad de Murcia, 30100 Espinardo, Spain
2. 

Real Observatorio de la Armada, ES-11 110 San Fernando

Received  September 2009 Revised  August 2010 Published  November 2010

The Hamilton-Jacobi equation in the sense of Poincaré, i.e. formulated in the extended phase space and including regularization, is revisited building canonical transformations with the purpose of Hamiltonian reduction and perturbation theory. We illustrate our approach dealing with attitude and orbital dynamics. Based on the use of Andoyer and Whittaker symplectic charts, for which all but one coordinates are cyclic in the Hamilton-Jacobi equation of the free rigid body motion and Kepler problem, respectively, we provide whole families of canonical transformations, among which one recognizes the familiar ones used in attitude and orbital dynamics. In addition, new canonical transformations are demonstrated.
Keywords: Poincaré regularization, Whittaker variables, Hamiltonian mechanics, Andoyer variables, symplectic transformations, rotational dynamics, Hamilton-Jacobi equation, orbital dynamics..
Mathematics Subject Classification: Primary: 34C20, 37N05, 47A55; Secondary: 70F15, 70E20, 70H2.
Citation: Sebastián Ferrer, Martin Lara. Families of canonical transformations by Hamilton-Jacobi-Poincaré equation. Application to rotational and orbital motion. Journal of Geometric Mechanics, 2010, 2 (3) : 223-241. doi: 10.3934/jgm.2010.2.223
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. Google Scholar

[2]

M. H. Andoyer, "Cours de Mécanique Céleste," 1, Gauthier-Villars et cie, Paris, 1923. (p 57) Google Scholar

[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.  Google Scholar

[4]

D. Boccaletti and G. Pucacco, "Theory of Orbits. 1: Integrable Systems and Non-perturbative Methods," Springer-Verlag, Berlin, 2001.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

R. H. Cushman and L. M. Bates, "Global Aspects of Classical Integrable Systems," Birkhäuser Verlag, Basel, 1997.  Google Scholar

[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) Google Scholar

[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.  Google Scholar

[10]

A. Deprit, The elimination of the parallax in satellite theory, Celes. Mech., 24 (1981), 111-153. doi: 10.1007/BF01229192.  Google Scholar

[11]

A. Deprit, A note concerning the TR-transformation, Celes. Mech., 23 (1981), 299-305. doi: 10.1007/BF01230743.  Google Scholar

[12]

A. Deprit and A. Elipe, Complete reduction of the Euler-Poinsot problem, Journal of the Astronautical Sciences, 41 (1993), 603-628.  Google Scholar

[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.  Google Scholar

[14]

F. Fassò, Superintegrable Hamiltonian systems: Geometry and perturbations, Acta Applicandae Mathematicae, 87 (2005), 93-121. doi: 10.1007/s10440-005-1139-8.  Google Scholar

[15]

S. Ferrer and M. Lara, "Orbital Canonical Transformations. The Isochronal Family," XII Jornadas Mecánica Celeste, Lalín, Spain, 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), 1899-1908. doi: 10.1088/0004-6256/139/5/1899.  Google Scholar

[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.  Google Scholar

[18]

H. Goldstein, C. Poole and J. Safko, "Classical Mechanics," 3rd edition, Pearson Addison-Wesley, 2002. Google Scholar

[19]

M. Hénon, L'amas isochrone I, Annales d'Astrophysique, 22 (1959), 126-139. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

J. V. José and E. J. Saletan, "Classical Dynamics: A Contemporary Approach," Cambridge University Press, 2000.  Google Scholar

[24]

H. Kinoshita, First-order perturbations of the two finite body problem, Publications of the Astronomical Society of Japan, 24 (1972), 423-457. Google Scholar

[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)  Google Scholar

[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.  Google Scholar

[27]

T. Levi-Civita, Nouvo sistema canonico di elementi ellittici, Annali di Matematica Serie III, XX (1913), 153-170. doi: 10.1007/BF02419588.  Google Scholar

[28]

T. Levi-Civita, Sur la régularization du problème des trois corps, Acta Mathematica, 42 (1918), 99-144. doi: 10.1007/BF02404404.  Google Scholar

[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.  Google Scholar

[30]

J. E. Marsden and T. Ratiu, "Introduction to Mechanics and Symmetry," TAM 17, 2nd edition, Springer, New York, 1999.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

J. Moser and E. J. Zehnder, "Notes on Dynamical Systems," Courant Lectures Notes in Mathematics, 12, AMS, Providence, 2005.  Google Scholar

[34]

J. P. Ortega and T. Ratiu, "Momentum Maps and Hamiltonian Reduction," Progress in Mathematics, 222. Birkhäuser, Boston, 2004.  Google Scholar

[35]

H. Poincaré, "Les Méthodes Mouvelles de la Mécanique Céleste," 2, Gauthier-Villars et Fils, Paris, 1893, pp. 315-342. Google Scholar

[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.] Google Scholar

[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. 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), 729-732.  Google Scholar

[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.  Google Scholar

[40]

G. J. Sussman, J. Wisdom and M. Mayer, "Structure and Interpretation of Classical Mechanics," The MIT Press, Cambridge, 2001.  Google Scholar

[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)  Google Scholar

[42]

P. Yanguas, Perturbations of the isochrone model, Nonlinearity, 14 (2001), 1-34. doi: 10.1088/0951-7715/14/1/301.  Google Scholar

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. Google Scholar

[2]

M. H. Andoyer, "Cours de Mécanique Céleste," 1, Gauthier-Villars et cie, Paris, 1923. (p 57) Google Scholar

[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.  Google Scholar

[4]

D. Boccaletti and G. Pucacco, "Theory of Orbits. 1: Integrable Systems and Non-perturbative Methods," Springer-Verlag, Berlin, 2001.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

R. H. Cushman and L. M. Bates, "Global Aspects of Classical Integrable Systems," Birkhäuser Verlag, Basel, 1997.  Google Scholar

[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) Google Scholar

[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.  Google Scholar

[10]

A. Deprit, The elimination of the parallax in satellite theory, Celes. Mech., 24 (1981), 111-153. doi: 10.1007/BF01229192.  Google Scholar

[11]

A. Deprit, A note concerning the TR-transformation, Celes. Mech., 23 (1981), 299-305. doi: 10.1007/BF01230743.  Google Scholar

[12]

A. Deprit and A. Elipe, Complete reduction of the Euler-Poinsot problem, Journal of the Astronautical Sciences, 41 (1993), 603-628.  Google Scholar

[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.  Google Scholar

[14]

F. Fassò, Superintegrable Hamiltonian systems: Geometry and perturbations, Acta Applicandae Mathematicae, 87 (2005), 93-121. doi: 10.1007/s10440-005-1139-8.  Google Scholar

[15]

S. Ferrer and M. Lara, "Orbital Canonical Transformations. The Isochronal Family," XII Jornadas Mecánica Celeste, Lalín, Spain, 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), 1899-1908. doi: 10.1088/0004-6256/139/5/1899.  Google Scholar

[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.  Google Scholar

[18]

H. Goldstein, C. Poole and J. Safko, "Classical Mechanics," 3rd edition, Pearson Addison-Wesley, 2002. Google Scholar

[19]

M. Hénon, L'amas isochrone I, Annales d'Astrophysique, 22 (1959), 126-139. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

J. V. José and E. J. Saletan, "Classical Dynamics: A Contemporary Approach," Cambridge University Press, 2000.  Google Scholar

[24]

H. Kinoshita, First-order perturbations of the two finite body problem, Publications of the Astronomical Society of Japan, 24 (1972), 423-457. Google Scholar

[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)  Google Scholar

[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.  Google Scholar

[27]

T. Levi-Civita, Nouvo sistema canonico di elementi ellittici, Annali di Matematica Serie III, XX (1913), 153-170. doi: 10.1007/BF02419588.  Google Scholar

[28]

T. Levi-Civita, Sur la régularization du problème des trois corps, Acta Mathematica, 42 (1918), 99-144. doi: 10.1007/BF02404404.  Google Scholar

[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.  Google Scholar

[30]

J. E. Marsden and T. Ratiu, "Introduction to Mechanics and Symmetry," TAM 17, 2nd edition, Springer, New York, 1999.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

J. Moser and E. J. Zehnder, "Notes on Dynamical Systems," Courant Lectures Notes in Mathematics, 12, AMS, Providence, 2005.  Google Scholar

[34]

J. P. Ortega and T. Ratiu, "Momentum Maps and Hamiltonian Reduction," Progress in Mathematics, 222. Birkhäuser, Boston, 2004.  Google Scholar

[35]

H. Poincaré, "Les Méthodes Mouvelles de la Mécanique Céleste," 2, Gauthier-Villars et Fils, Paris, 1893, pp. 315-342. Google Scholar

[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.] Google Scholar

[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. 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), 729-732.  Google Scholar

[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.  Google Scholar

[40]

G. J. Sussman, J. Wisdom and M. Mayer, "Structure and Interpretation of Classical Mechanics," The MIT Press, Cambridge, 2001.  Google Scholar

[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)  Google Scholar

[42]

P. Yanguas, Perturbations of the isochrone model, Nonlinearity, 14 (2001), 1-34. doi: 10.1088/0951-7715/14/1/301.  Google Scholar

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