American Institute of Mathematical Sciences

Advanced
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

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.  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

show all references

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

[1]

Alicia Cordero, José Martínez Alfaro, Pura Vindel. Bott integrable Hamiltonian systems on $S^{2}\times S^{1}$. Discrete & 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 & 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 & 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 & 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 & 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 & 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 & 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 & 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 & 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 & 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 & 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 & 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 & Continuous Dynamical Systems, 2013, 33 (9) : 4123-4155. doi: 10.3934/dcds.2013.33.4123
[16]

Kai Koike. Wall effect on the motion of a rigid body immersed in a free molecular flow. Kinetic & Related Models, 2018, 11 (3) : 441-467. doi: 10.3934/krm.2018020
[17]

Matthias Morzfeld, Daniel T. Kawano, Fai Ma. Characterization of damped linear dynamical systems in free motion. Numerical Algebra, Control & Optimization, 2013, 3 (1) : 49-62. doi: 10.3934/naco.2013.3.49
[18]

Jeremias Epperlein, Stefan Siegmund. Phase-locked trajectories for dynamical systems on graphs. Discrete & 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 & Continuous Dynamical Systems, 2006, 16 (3) : 635-655. doi: 10.3934/dcds.2006.16.635
[20]

Peter Bella, Arianna Giunti. Green's function for elliptic systems: Moment bounds. Networks & Heterogeneous Media, 2018, 13 (1) : 155-176. doi: 10.3934/nhm.2018007

2020 Impact Factor: 0.857

Tools

Metrics

  • PDF downloads (216)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy