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Aspects of reduction and transformation of Lagrangian systems with symmetry
Andoyer's variables and phases in the free rigid body
1. | Departamento de Matemática Aplicada, Universidad de Murcia, 30100 Espinardo |
2. | Departamento de Matemática Aplicada, Universidad de Murcia, Murcia, 30071 Espinardo, Spain |
References:
[1] |
M. H. Andoyer, Cours de Mécanique Céleste, The Mathematical Gazette, 12 (1924), p. 30.
doi: 10.2307/3603410. |
[2] |
L. Bates, R. Cushman and E. Savev, The rotation number and the herpolhode angle in Eulers top, Z. angew. Math. Phys., 56 (2005), 183-191.
doi: 10.1007/s00033-004-2082-7. |
[3] |
A. V. Borisov, A. A. Kilin and I. S. Mamaev, Absolute and Relative Choreographies in Rigid Body Dynamics, Regular and Chaotic Dynamics, 13 (2008), 204-220. Available from: http://ics.org.ru/doc?pdf=1279&dir=e
doi: 10.1134/S1560354708030064. |
[4] |
R. Cushman and L. Bates, Global Aspects of Classical Integrable Systems, Birhäuser Verlag, Basel, 1997.
doi: 10.1007/978-3-0348-8891-2. |
[5] |
A. Deprit, Free rotation of a rigid body studied in the phase space, American Journal of Physics, 35 (1967), 424-428. |
[6] |
F. Fassò, The EulerPoinsot top: A non-commutatively integrable system without global action-angle coordinates, J. Appl. Math. Phys. (ZAMP), 47 (1996), 953-976.
doi: 10.1007/BF00920045. |
[7] |
T. Fukushima, Precise and fast computation of a general incomplete elliptic integral of third kind by half and double argument transformations, Journal of Computational and Applied Mathematics, 236 (2012), 1961-1975.
doi: 10.1016/j.cam.2011.11.007. |
[8] |
W. B. Heard, Rigid Body Mechanics. Mathematics, Physics and Applications, WILEY-VCH Verlag GmbH&Co. KGaA, Weinheim, 2006. |
[9] |
D. D. Holm and J. E. Marsden, The rotor and the pendulum, in Symplectic Geometry and Mathematical Physics. Actes du colloque en l'honneur de Jean-Marie Souriau, Ed. by P. Donato et al., Prog. in Math., 99 (1991), Birkhäuser, 189-203. Available from: http://www.cds.caltech.edu/~marsden/bib/1991/06-HoMa1991/HoMa1991.pdf. |
[10] |
D. F. Lawden, Elliptic Functions and Applications, Springer-Verlag, New York, 1989. |
[11] |
M. Levi, Geometric phases in the motion of rigid bodies, Archive for Rational Mechanics and Analysis, 122 (1993), 213-229.
doi: 10.1007/BF00380255. |
[12] |
M. Levi, Lectures on geometrical methods in mechanics, in Classical and Celestial Mechanics, 239-280, Princeton Univ. Press, Princeton, NJ, 2002. |
[13] |
J. E. Marsden, Geometric foundations of motion and control, in Motion, Control, and Geometry: Proceedings of a Symposium, Nonlinear Sci. Today 1996, 21 pp. |
[14] |
J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry, Texts in Applied Mathematics, Springer, New York, Second Ed., 1999.
doi: 10.1007/978-0-387-21792-5. |
[15] |
R. Montgomery, How much does the rigid body rotate? A Berry's phase from the $18^{th}$ century, American Journal of Physics, 59 (1991), 394-398. Available from: http://montgomery.math.ucsc.edu/papers/rigid_body.pdf
doi: 10.1119/1.16514. |
[16] |
R. Natário, An elementary derivation of the Montgomery phase formula for the Euler Top, Journal of Geometric Mechanics, 2 (2010), 113-118. arXiv:0909.2109v3
doi: 10.3934/jgm.2010.2.113. |
[17] |
J. P. Ortega and T. S. Ratiu, Momentum Maps and Hamiltonian Reduction, Progress in Mathematics, 222. Birkhäuser Boston, Inc., Boston, MA, 2004. |
[18] |
P. Tantalo, Geometric Phases for the Free Rigid Body with Variable Inertia Tensor, Ph.D thesis, University of California in Santa Cruz, 1993. |
[19] |
S. Wolfram, Wolfram Mathematica 9, Wolfram Research Inc./Cambridge Univ. Press, Cambridge, 2003, http://reference.wolfram.com/mathematica/guide/Mathematica.html. |
[20] |
V. F. Zhuravlev, The solid angle theorem in rigid body dynamics, J. Appl. Maths. Mechs., 60 (1996), 319-322.
doi: 10.1016/0021-8928(96)00040-8. |
show all references
References:
[1] |
M. H. Andoyer, Cours de Mécanique Céleste, The Mathematical Gazette, 12 (1924), p. 30.
doi: 10.2307/3603410. |
[2] |
L. Bates, R. Cushman and E. Savev, The rotation number and the herpolhode angle in Eulers top, Z. angew. Math. Phys., 56 (2005), 183-191.
doi: 10.1007/s00033-004-2082-7. |
[3] |
A. V. Borisov, A. A. Kilin and I. S. Mamaev, Absolute and Relative Choreographies in Rigid Body Dynamics, Regular and Chaotic Dynamics, 13 (2008), 204-220. Available from: http://ics.org.ru/doc?pdf=1279&dir=e
doi: 10.1134/S1560354708030064. |
[4] |
R. Cushman and L. Bates, Global Aspects of Classical Integrable Systems, Birhäuser Verlag, Basel, 1997.
doi: 10.1007/978-3-0348-8891-2. |
[5] |
A. Deprit, Free rotation of a rigid body studied in the phase space, American Journal of Physics, 35 (1967), 424-428. |
[6] |
F. Fassò, The EulerPoinsot top: A non-commutatively integrable system without global action-angle coordinates, J. Appl. Math. Phys. (ZAMP), 47 (1996), 953-976.
doi: 10.1007/BF00920045. |
[7] |
T. Fukushima, Precise and fast computation of a general incomplete elliptic integral of third kind by half and double argument transformations, Journal of Computational and Applied Mathematics, 236 (2012), 1961-1975.
doi: 10.1016/j.cam.2011.11.007. |
[8] |
W. B. Heard, Rigid Body Mechanics. Mathematics, Physics and Applications, WILEY-VCH Verlag GmbH&Co. KGaA, Weinheim, 2006. |
[9] |
D. D. Holm and J. E. Marsden, The rotor and the pendulum, in Symplectic Geometry and Mathematical Physics. Actes du colloque en l'honneur de Jean-Marie Souriau, Ed. by P. Donato et al., Prog. in Math., 99 (1991), Birkhäuser, 189-203. Available from: http://www.cds.caltech.edu/~marsden/bib/1991/06-HoMa1991/HoMa1991.pdf. |
[10] |
D. F. Lawden, Elliptic Functions and Applications, Springer-Verlag, New York, 1989. |
[11] |
M. Levi, Geometric phases in the motion of rigid bodies, Archive for Rational Mechanics and Analysis, 122 (1993), 213-229.
doi: 10.1007/BF00380255. |
[12] |
M. Levi, Lectures on geometrical methods in mechanics, in Classical and Celestial Mechanics, 239-280, Princeton Univ. Press, Princeton, NJ, 2002. |
[13] |
J. E. Marsden, Geometric foundations of motion and control, in Motion, Control, and Geometry: Proceedings of a Symposium, Nonlinear Sci. Today 1996, 21 pp. |
[14] |
J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry, Texts in Applied Mathematics, Springer, New York, Second Ed., 1999.
doi: 10.1007/978-0-387-21792-5. |
[15] |
R. Montgomery, How much does the rigid body rotate? A Berry's phase from the $18^{th}$ century, American Journal of Physics, 59 (1991), 394-398. Available from: http://montgomery.math.ucsc.edu/papers/rigid_body.pdf
doi: 10.1119/1.16514. |
[16] |
R. Natário, An elementary derivation of the Montgomery phase formula for the Euler Top, Journal of Geometric Mechanics, 2 (2010), 113-118. arXiv:0909.2109v3
doi: 10.3934/jgm.2010.2.113. |
[17] |
J. P. Ortega and T. S. Ratiu, Momentum Maps and Hamiltonian Reduction, Progress in Mathematics, 222. Birkhäuser Boston, Inc., Boston, MA, 2004. |
[18] |
P. Tantalo, Geometric Phases for the Free Rigid Body with Variable Inertia Tensor, Ph.D thesis, University of California in Santa Cruz, 1993. |
[19] |
S. Wolfram, Wolfram Mathematica 9, Wolfram Research Inc./Cambridge Univ. Press, Cambridge, 2003, http://reference.wolfram.com/mathematica/guide/Mathematica.html. |
[20] |
V. F. Zhuravlev, The solid angle theorem in rigid body dynamics, J. Appl. Maths. Mechs., 60 (1996), 319-322.
doi: 10.1016/0021-8928(96)00040-8. |
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