# American Institute of Mathematical Sciences

March  2014, 6(1): 25-37. doi: 10.3934/jgm.2014.6.25

## Andoyer's variables and phases in the free rigid body

Received  May 2013 Revised  November 2013 Published  April 2014

Using Andoyer's variables we present a new proof of Montgomery's formula by measuring $\Delta\mu$ when $\nu$ has made a rotation. Our treatment is built on the equations of the differential system of the free rigid solid, together with the explicit expression of the spherical area defined by the intersection of the surfaces given by the energy and momentum integrals. We also consider the phase $\Delta\nu$ of the moving frame when $\mu$ has made a rotation around the angular momentum vector, and we give the formula for its computation.
Citation: Sebastián Ferrer, Francisco J. Molero. Andoyer's variables and phases in the free rigid body. Journal of Geometric Mechanics, 2014, 6 (1) : 25-37. doi: 10.3934/jgm.2014.6.25
##### References:
 [1] M. H. Andoyer, Cours de Mécanique Céleste, The Mathematical Gazette, 12 (1924), p. 30. doi: 10.2307/3603410.  Google Scholar [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.  Google Scholar [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.  Google Scholar [4] R. Cushman and L. Bates, Global Aspects of Classical Integrable Systems, Birhäuser Verlag, Basel, 1997. doi: 10.1007/978-3-0348-8891-2.  Google Scholar [5] A. Deprit, Free rotation of a rigid body studied in the phase space, American Journal of Physics, 35 (1967), 424-428. Google Scholar [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.  Google Scholar [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.  Google Scholar [8] W. B. Heard, Rigid Body Mechanics. Mathematics, Physics and Applications, WILEY-VCH Verlag GmbH&Co. KGaA, Weinheim, 2006. Google Scholar [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.  Google Scholar [10] D. F. Lawden, Elliptic Functions and Applications, Springer-Verlag, New York, 1989.  Google Scholar [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.  Google Scholar [12] M. Levi, Lectures on geometrical methods in mechanics, in Classical and Celestial Mechanics, 239-280, Princeton Univ. Press, Princeton, NJ, 2002.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [17] J. P. Ortega and T. S. Ratiu, Momentum Maps and Hamiltonian Reduction, Progress in Mathematics, 222. Birkhäuser Boston, Inc., Boston, MA, 2004.  Google Scholar [18] P. Tantalo, Geometric Phases for the Free Rigid Body with Variable Inertia Tensor, Ph.D thesis, University of California in Santa Cruz, 1993.  Google Scholar [19] S. Wolfram, Wolfram Mathematica 9, Wolfram Research Inc./Cambridge Univ. Press, Cambridge, 2003, http://reference.wolfram.com/mathematica/guide/Mathematica.html. Google Scholar [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.  Google Scholar

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##### References:
 [1] M. H. Andoyer, Cours de Mécanique Céleste, The Mathematical Gazette, 12 (1924), p. 30. doi: 10.2307/3603410.  Google Scholar [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.  Google Scholar [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.  Google Scholar [4] R. Cushman and L. Bates, Global Aspects of Classical Integrable Systems, Birhäuser Verlag, Basel, 1997. doi: 10.1007/978-3-0348-8891-2.  Google Scholar [5] A. Deprit, Free rotation of a rigid body studied in the phase space, American Journal of Physics, 35 (1967), 424-428. Google Scholar [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.  Google Scholar [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.  Google Scholar [8] W. B. Heard, Rigid Body Mechanics. Mathematics, Physics and Applications, WILEY-VCH Verlag GmbH&Co. KGaA, Weinheim, 2006. Google Scholar [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.  Google Scholar [10] D. F. Lawden, Elliptic Functions and Applications, Springer-Verlag, New York, 1989.  Google Scholar [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.  Google Scholar [12] M. Levi, Lectures on geometrical methods in mechanics, in Classical and Celestial Mechanics, 239-280, Princeton Univ. Press, Princeton, NJ, 2002.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [17] J. P. Ortega and T. S. Ratiu, Momentum Maps and Hamiltonian Reduction, Progress in Mathematics, 222. Birkhäuser Boston, Inc., Boston, MA, 2004.  Google Scholar [18] P. Tantalo, Geometric Phases for the Free Rigid Body with Variable Inertia Tensor, Ph.D thesis, University of California in Santa Cruz, 1993.  Google Scholar [19] S. Wolfram, Wolfram Mathematica 9, Wolfram Research Inc./Cambridge Univ. Press, Cambridge, 2003, http://reference.wolfram.com/mathematica/guide/Mathematica.html. Google Scholar [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.  Google Scholar
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