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

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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

Abstract
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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 and 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.
[2]	L. Bates and F. Fassò, The conjugate locus for the euler top. I. The axisymmetric case, Int. Math. Forum, 2 (2007), 2109-2139.
[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.
[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.
[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.
[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.
[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.
[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.
[9]	V. Jurdjevic, "Geometric Control Theory,'' Cambridge studies in advanced mathematics, Cambridge University Press, Cambridge, 1997.
[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.
[11]	D. Lawden, "Elliptic Functions and Applications,'' Applied mathematical sciences, Springer-Verlag, New York, 1989, xiv+334.
[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.
[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.
[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.
[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.
[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.

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.
[2]	L. Bates and F. Fassò, The conjugate locus for the euler top. I. The axisymmetric case, Int. Math. Forum, 2 (2007), 2109-2139.
[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.
[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.
[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.
[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.
[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.
[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.
[9]	V. Jurdjevic, "Geometric Control Theory,'' Cambridge studies in advanced mathematics, Cambridge University Press, Cambridge, 1997.
[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.
[11]	D. Lawden, "Elliptic Functions and Applications,'' Applied mathematical sciences, Springer-Verlag, New York, 1989, xiv+334.
[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.
[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.
[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.
[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.
[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.

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