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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.

[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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