Andoyer's variables and phases in the free rigid body

Previous Article
Fluid-structure interaction in the Lagrange-Poincaré formalism: The Navier-Stokes and inviscid regimes
JGM Home
This Issue
Next Article
Aspects of reduction and transformation of Lagrangian systems with symmetry

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

Andoyer's variables and phases in the free rigid body

Sebastián Ferrer ^1, and Francisco J. Molero ^2,

1.	Departamento de Matemática Aplicada, Universidad de Murcia, 30100 Espinardo
2.	Departamento de Matemática Aplicada, Universidad de Murcia, Murcia, 30071 Espinardo, Spain

Received May 2013 Revised November 2013 Published April 2014

Abstract
Related Papers

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.

Keywords: integrable Hamiltonian systems, Montgomery's phase, free rigid body, Andoyer's variables., Dynamical systems.

Mathematics Subject Classification: Primary: 70E1.

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.

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.

[1]	Alicia Cordero, José Martínez Alfaro, Pura Vindel. Bott integrable Hamiltonian systems on $S^{2}\times S^{1}$. Discrete and Continuous Dynamical Systems, 2008, 22 (3) : 587-604. doi: 10.3934/dcds.2008.22.587
[2]	Kuo-Chang Chen. On Chenciner-Montgomery's orbit in the three-body problem. Discrete and Continuous Dynamical Systems, 2001, 7 (1) : 85-90. doi: 10.3934/dcds.2001.7.85
[3]	Giovanni Forni, Howard Masur, John Smillie. Bill Veech's contributions to dynamical systems. Journal of Modern Dynamics, 2019, 14: v-xxv. doi: 10.3934/jmd.2019v
[4]	Sonja Hohloch, Silvia Sabatini, Daniele Sepe. From compact semi-toric systems to Hamiltonian $S^1$-spaces. Discrete and Continuous Dynamical Systems, 2015, 35 (1) : 247-281. doi: 10.3934/dcds.2015.35.247
[5]	Ernest Fontich, Pau Martín. Arnold diffusion in perturbations of analytic integrable Hamiltonian systems. Discrete and Continuous Dynamical Systems, 2001, 7 (1) : 61-84. doi: 10.3934/dcds.2001.7.61
[6]	H. M. Hastings, S. Silberger, M. T. Weiss, Y. Wu. A twisted tensor product on symbolic dynamical systems and the Ashley's problem. Discrete and Continuous Dynamical Systems, 2003, 9 (3) : 549-558. doi: 10.3934/dcds.2003.9.549
[7]	Arno Berger. Multi-dimensional dynamical systems and Benford's Law. Discrete and Continuous Dynamical Systems, 2005, 13 (1) : 219-237. doi: 10.3934/dcds.2005.13.219
[8]	Bernard Bonnard, Olivier Cots, Nataliya Shcherbakova. The Serret-Andoyer Riemannian metric and Euler-Poinsot rigid body motion. Mathematical Control and Related Fields, 2013, 3 (3) : 287-302. doi: 10.3934/mcrf.2013.3.287
[9]	Sebastián Ferrer, Francisco Crespo. Parametric quartic Hamiltonian model. A unified treatment of classic integrable systems. Journal of Geometric Mechanics, 2014, 6 (4) : 479-502. doi: 10.3934/jgm.2014.6.479
[10]	Fuzhong Cong, Jialin Hong, Hongtian Li. Quasi-effective stability for nearly integrable Hamiltonian systems. Discrete and Continuous Dynamical Systems - B, 2016, 21 (1) : 67-80. doi: 10.3934/dcdsb.2016.21.67
[11]	Hahng-Yun Chu, Se-Hyun Ku, Jong-Suh Park. Conley's theorem for dispersive systems. Discrete and Continuous Dynamical Systems - S, 2015, 8 (2) : 313-321. doi: 10.3934/dcdss.2015.8.313
[12]	Abed Bounemoura, Edouard Pennamen. Instability for a priori unstable Hamiltonian systems: A dynamical approach. Discrete and Continuous Dynamical Systems, 2012, 32 (3) : 753-793. doi: 10.3934/dcds.2012.32.753
[13]	Mikhail B. Sevryuk. Invariant tori in quasi-periodic non-autonomous dynamical systems via Herman's method. Discrete and Continuous Dynamical Systems, 2007, 18 (2&3) : 569-595. doi: 10.3934/dcds.2007.18.569
[14]	David Cheban. I. U. Bronshtein's conjecture for monotone nonautonomous dynamical systems. Discrete and Continuous Dynamical Systems - B, 2019, 24 (3) : 1095-1113. doi: 10.3934/dcdsb.2019008
[15]	Zhiming Li, Lin Shu. The metric entropy of random dynamical systems in a Hilbert space: Characterization of invariant measures satisfying Pesin's entropy formula. Discrete and Continuous Dynamical Systems, 2013, 33 (9) : 4123-4155. doi: 10.3934/dcds.2013.33.4123
[16]	Matthias Morzfeld, Daniel T. Kawano, Fai Ma. Characterization of damped linear dynamical systems in free motion. Numerical Algebra, Control and Optimization, 2013, 3 (1) : 49-62. doi: 10.3934/naco.2013.3.49
[17]	Kai Koike. Wall effect on the motion of a rigid body immersed in a free molecular flow. Kinetic and Related Models, 2018, 11 (3) : 441-467. doi: 10.3934/krm.2018020
[18]	Jeremias Epperlein, Stefan Siegmund. Phase-locked trajectories for dynamical systems on graphs. Discrete and Continuous Dynamical Systems - B, 2013, 18 (7) : 1827-1844. doi: 10.3934/dcdsb.2013.18.1827
[19]	Dongfeng Zhang, Junxiang Xu. On elliptic lower dimensional tori for Gevrey-smooth Hamiltonian systems under Rüssmann's non-degeneracy condition. Discrete and Continuous Dynamical Systems, 2006, 16 (3) : 635-655. doi: 10.3934/dcds.2006.16.635
[20]	Marcel Guardia. Splitting of separatrices in the resonances of nearly integrable Hamiltonian systems of one and a half degrees of freedom. Discrete and Continuous Dynamical Systems, 2013, 33 (7) : 2829-2859. doi: 10.3934/dcds.2013.33.2829

2021 Impact Factor: 0.737

Tools

Metrics

Other articles
by authors

on AIMS
Sebastián Ferrer Francisco J. Molero
on Google Scholar
Sebastián Ferrer Francisco J. Molero

[Back to Top]