# 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. [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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##### 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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