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September  2013, 3(3): 287-302. doi: 10.3934/mcrf.2013.3.287

The Serret-Andoyer Riemannian metric and Euler-Poinsot rigid body motion

Bernard Bonnard 1, , Olivier Cots 2, and  Nataliya Shcherbakova 2,

1. 

Institut de mathématiques de Bourgogne, 9 avenue Savary, 21078 Dijon, France
2. 

INRIA Sophia Antipolis Méditerranée, B.P. 93, route des Lucioles, 06902 Sophia Antipolis, France, France

Received  December 2012 Revised  March 2013 Published  September 2013

The Euler-Poinsot rigid body motion is a standard mechanical system and is the model for left-invariant Riemannian metrics on $SO(3)$. In this article, using the Serret-Andoyer variables we parameterize the solutions and compute the Jacobi fields in relation with the conjugate locus evaluation. Moreover the metric can be restricted to a 2D surface and the conjugate points of this metric are evaluated using recent work [4] on surfaces of revolution.
Keywords: Euler-Poinsot rigid body motion, conjugate locus on surfaces of revolution..
Mathematics Subject Classification: 49K15, 53C20, 70Q0.
Citation: 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
References:
[1]

V. I. Arnold, "Mathematical Methods of Classical Mechanics," Translated from the Russian by K. Vogtmann and A. Weinstein, $2^{nd}$ edition, Graduate Texts in Mathematics, 60. Springer-Verlag, New York, 1989, xvi+508.  Google Scholar

[2]

L. Bates and F. Fassò, The conjugate locus for the euler top. I. The axisymmetric case, Int. Math. Forum, 2 (2007), 2109-2139.  Google Scholar

[3]

B. Bonnard, Contrôlabilité de systemes mécaniques sur les groupes de lie (French), [controllability of mechanical systems on lie groups], SIAM J. Control Optim., 22 (1984), 711-722. doi: 10.1137/0322045.  Google Scholar

[4]

B. Bonnard, J.-B. Caillau, R. Sinclair and M. Tanaka, Conjugate and cut loci of a two-sphere of revolution with application to optimal control, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 1081-1098. doi: 10.1016/j.anihpc.2008.03.010.  Google Scholar

[5]

B. Bonnard, O. Cots and N. Shcherbakova, Energy minimization problem in two-level dissipative quantum control: Meridian case, J. Math. Sci., (2013) to appear. Google Scholar

[6]

M. do Carmo, "Riemannian Geometry,'' Translated from the second Portuguese edition by Francis Flaherty, Mathematics: Theory & Applications. Birkhäuser Boston, Inc., Boston, MA, 1992, xiv+300.  Google Scholar

[7]

P. Gurfil, A. Elipe, W. Tangren and M. Efroimsky, The Serret-Andoyer formalism in rigid-body dynamics. I. Symmetries and perturbations, Regul. Chaotic Dyn., 12 (2007), 389-425. doi: 10.1134/S156035470704003X.  Google Scholar

[8]

J. Itoh and K. Kiyohara, The cut loci and the conjugate loci on ellipsoids, Manuscripta Math., 114 (2004), 247-264. doi: 10.1007/s00229-004-0455-z.  Google Scholar

[9]

V. Jurdjevic, "Geometric Control Theory,'' Cambridge studies in advanced mathematics, Cambridge University Press, Cambridge, 1997.  Google Scholar

[10]

M. Lara and S. Ferrer, Closed form integration of the Hitzl-Breakwell problem in action-angle variables,, IAA-AAS-DyCoSS1-01-02 (AAS 12-302), (): 1.   Google Scholar

[11]

D. Lawden, "Elliptic Functions and Applications,'' Applied mathematical sciences, Springer-Verlag, New York, 1989, xiv+334.  Google Scholar

[12]

H. Poincaré, Sur les lignes géodésiques des surfaces convexes, (French) [On the geodesic lines of convex surfaces] Trans. Amer. Math. Soc., 6 (1905), 237-274. doi: 10.2307/1986219.  Google Scholar

[13]

K. Shiohama, T. Shioya and M. Tanaka, "The Geometry of Total Curvature on Complete Open Surfaces,'' Cambridge tracts in mathematics, 159. Cambridge University Press, Cambridge, 2003, x+284. doi: 10.1017/CBO9780511543159.  Google Scholar

[14]

R. Sinclair and M. Tanaka, The cut locus of a two-sphere of revolution and toponogov's comparison theorem, Tohoku Math. J. (2), 59 (2007), 379-399. doi: 10.2748/tmj/1192117984.  Google Scholar

[15]

A. M. Vershik and V. Ya. Gershkovich, Nonholonomic dynamical systems, geometry of distributions, and variational problems, vol. 16 of Dynamical Systems VII, Encyclopaedia of Mathematical Sciences, Springer-Verlag, Berlin-Heidelberg-New York, 1994, 1-81. Google Scholar

[16]

H. Yuan, R. Zeier, N. Khaneja and S. Lloyd, Constructing two-qubit gates with minimal couplings, Phys. Rev. A (3), 79 (2009), 4pp. doi: 10.1103/PhysRevA.79.042309.  Google Scholar

show all references

References:
[1]

V. I. Arnold, "Mathematical Methods of Classical Mechanics," Translated from the Russian by K. Vogtmann and A. Weinstein, $2^{nd}$ edition, Graduate Texts in Mathematics, 60. Springer-Verlag, New York, 1989, xvi+508.  Google Scholar

[2]

L. Bates and F. Fassò, The conjugate locus for the euler top. I. The axisymmetric case, Int. Math. Forum, 2 (2007), 2109-2139.  Google Scholar

[3]

B. Bonnard, Contrôlabilité de systemes mécaniques sur les groupes de lie (French), [controllability of mechanical systems on lie groups], SIAM J. Control Optim., 22 (1984), 711-722. doi: 10.1137/0322045.  Google Scholar

[4]

B. Bonnard, J.-B. Caillau, R. Sinclair and M. Tanaka, Conjugate and cut loci of a two-sphere of revolution with application to optimal control, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 1081-1098. doi: 10.1016/j.anihpc.2008.03.010.  Google Scholar

[5]

B. Bonnard, O. Cots and N. Shcherbakova, Energy minimization problem in two-level dissipative quantum control: Meridian case, J. Math. Sci., (2013) to appear. Google Scholar

[6]

M. do Carmo, "Riemannian Geometry,'' Translated from the second Portuguese edition by Francis Flaherty, Mathematics: Theory & Applications. Birkhäuser Boston, Inc., Boston, MA, 1992, xiv+300.  Google Scholar

[7]

P. Gurfil, A. Elipe, W. Tangren and M. Efroimsky, The Serret-Andoyer formalism in rigid-body dynamics. I. Symmetries and perturbations, Regul. Chaotic Dyn., 12 (2007), 389-425. doi: 10.1134/S156035470704003X.  Google Scholar

[8]

J. Itoh and K. Kiyohara, The cut loci and the conjugate loci on ellipsoids, Manuscripta Math., 114 (2004), 247-264. doi: 10.1007/s00229-004-0455-z.  Google Scholar

[9]

V. Jurdjevic, "Geometric Control Theory,'' Cambridge studies in advanced mathematics, Cambridge University Press, Cambridge, 1997.  Google Scholar

[10]

M. Lara and S. Ferrer, Closed form integration of the Hitzl-Breakwell problem in action-angle variables,, IAA-AAS-DyCoSS1-01-02 (AAS 12-302), (): 1.   Google Scholar

[11]

D. Lawden, "Elliptic Functions and Applications,'' Applied mathematical sciences, Springer-Verlag, New York, 1989, xiv+334.  Google Scholar

[12]

H. Poincaré, Sur les lignes géodésiques des surfaces convexes, (French) [On the geodesic lines of convex surfaces] Trans. Amer. Math. Soc., 6 (1905), 237-274. doi: 10.2307/1986219.  Google Scholar

[13]

K. Shiohama, T. Shioya and M. Tanaka, "The Geometry of Total Curvature on Complete Open Surfaces,'' Cambridge tracts in mathematics, 159. Cambridge University Press, Cambridge, 2003, x+284. doi: 10.1017/CBO9780511543159.  Google Scholar

[14]

R. Sinclair and M. Tanaka, The cut locus of a two-sphere of revolution and toponogov's comparison theorem, Tohoku Math. J. (2), 59 (2007), 379-399. doi: 10.2748/tmj/1192117984.  Google Scholar

[15]

A. M. Vershik and V. Ya. Gershkovich, Nonholonomic dynamical systems, geometry of distributions, and variational problems, vol. 16 of Dynamical Systems VII, Encyclopaedia of Mathematical Sciences, Springer-Verlag, Berlin-Heidelberg-New York, 1994, 1-81. Google Scholar

[16]

H. Yuan, R. Zeier, N. Khaneja and S. Lloyd, Constructing two-qubit gates with minimal couplings, Phys. Rev. A (3), 79 (2009), 4pp. doi: 10.1103/PhysRevA.79.042309.  Google Scholar

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