The discriminant set and bifurcation diagram of the integrable case of M. Adler and P. van Moerbeke

P. E. Ryabov ; E. O. Birucheva

Russian journal of nonlinear dynamics, Tome 12 (2016) no. 4, pp. 633-650.

Résumé

The paper presents explicitly the spectral curve and the discriminant set of the integrable case of M. Adler and P. van Moerbeke. For critical points of rank 0 and 1 of the momentum map we explicitly calculate the characteristic values defining their type. An algorithm is proposed for finding the bifurcation diagram from the real part of the discriminant set with the help of critical points of rank 0 and 1. The algorithm works under the condition that the real part of the discriminant set contains the bifurcation diagram.

Détail

Keywords: integrable Hamiltonian systems, spectral curve, bifurcation diagram.
Mots-clés : discriminant set

Bibliographie

[1] Adler M., van Moerbeke P. A., “A new geodesic flow on $so(4)$”, Probability, Statistical Mechanics and Number Theory: A Volume Dedicated to Mark Kac, Adv. Math. Suppl. Stud., 9, ed. G. -C. Rota, Acad. Press, Orlando, Fla., 1986, 81–96

[2] Mishchenko A. S., Fomenko A. T., “Euler equations on finite-dimensional Lie groups”, Math. USSR-Izv., 12:2 (1978), 371–389 | DOI | MR | Zbl

[3] Mishchenko A. S., Fomenko A. T., “Integrability of Euler's equations on semisimple Lie algebras”, Tr. Semin. Vektorn. Tenzorn. Anal., 19 (1979), 3–94 (Russian)

[4] Borisov A. V., Mamaev I. S., Sokolov V. V., “A new integrable case on $so(4)$”, Dokl. Phys., 46:12 (2001), 888–889 | DOI | MR

[5] Greenhill A. G., “On the general motion of a liquid ellipsoid under the gravitation of its own parts”, Proc. Cambridge Philos. Soc., 4 (1880), 4–14 | Zbl

[6] Zhukovskii N. E., “Motion of a rigid body containing a cavity filled with a homogeneous continuous liquid”: Zhukovskii N. E., Collected Works, v. 1, Gostekhteorizdat, Moscow, 1949, 31–152 (Russian)

[7] Poincaré H., “Sur la précession des corps déformables”, Bull. Astron., 27 (1910), 321–356 | Zbl

[8] Moiseyev N. N., Rumyantsev V. V., Dynamic stability of bodies containing fluid, Springer, New York, 1968, XVI, 346 pp. | Zbl

[9] Stekloff V. A., “Sur le movement d’un corps solide ayant une cavité de forme ellipsoidale remplie par un liquide incompressible et sur les variations des latitudes”, Ann. Fac. Sci. Toulouse Math. (3), 1 (1909), 145–256 | DOI | MR

[10] Fomenko A. T., Symplectic geometry, Adv. Stud. Contemp. Math., 5, 2nd ed., CRC Press, Boca Raton, Fla., 1995, 484 pp. | MR

[11] Adler M., van Moerbeke P., Vanhaecke P., Algebraic integrability, Painlevé geometry and Lie algebras, Ergeb. Math. Grenzgeb. (3), 47, Springer, Berlin, 2004, 483 pp. | MR

[12] Kozlov V. V., Symmetries, topology and resonances in Hamiltonian mechanics, Ergeb. Math. Grenzgeb. (3), 31, Springer, Berlin, 1996, xii+378 pp. | MR | MR

[13] Borisov A. V., Mamaev I. S., Modern methods of the theory of integrable systems, R Dynamics, ICS, Moscow, 2003 (Russian) | MR

[14] Borisov A. V., Mamaev I. S., Dynamics of a rigid body: Hamiltonian methods, integrability, chaos, 2nd ed., R Dynamics, Institute of Computer Science, Izhevsk, 2005, 576 pp. (Russian) | MR

[15] Bogoyavlenskii O. I., Overturning solitons. Nonlinear integrable equations, Nauka, Moscow, 1991 (Russian) | MR

[16] Audin M., Spinning tops: A course on integrable systems, Cambridge Stud. Adv. Math., 51, Cambridge Univ. Press, Cambridge, 1999, 148 pp. | MR | Zbl

[17] Brailov Yu. A., “Geometry of translations of invariants on semisimple Lie algebras”, Sb. Math., 194:11–12 (2003), 1585–1598 | DOI | DOI | MR | Zbl

[18] Ryabov P. E., “Algebraic curves and bifurcation diagrams of two integrable problems”, Mekh. Tverd. Tela, 2007, no. 37, 97–111 (Russian)

[19] Bolsinov A. V., Oshemkov A. A., “Bi-Hamiltonian structures and singularities of integrable systems”, Regul. Chaotic Dyn., 14:4–5 (2009), 431–454 | DOI | MR | Zbl

[20] Konyaev A. Yu., “The bifurcation diagram and discriminant of a spectral curve of integrable systems on Lie algebras”, Sb. Math., 201:9–10 (2010), 1273–1305 | DOI | DOI | MR | Zbl

[21] Bolsinov A., Izosimov A., “Singularities of Bi-Hamiltonian systems”, Comm. Math. Phys., 331:2 (2014), 507–543 | DOI | MR | Zbl

[22] Ryabov P. E., “New invariant relations for the generalized two-field gyrostat”, J. Geom. Phys., 87 (2015), 415–421 | DOI | MR | Zbl

[23] Izosimov A., “Singularities of integrable systems and algebraic curves”, Int. Math. Res. Notices, 2016:17 (2016), 50 pp.

[24] Reyman A. G., Semenov-Tian-Shansky M. A., “A new integrable case of the motion of the $4$-dimensional rigid body”, Comm. Math. Phys., 105:3 (1986), 461–472 | DOI | MR | Zbl

[25] Bolsinov A. V., Borisov A. V., “Compatible Poisson brackets on Lie algebras”, Math. Notes, 72:1–2 (2002), 10–30 | DOI | DOI | MR | Zbl

[26] Malkin I. G., Theory of stability of motion, Univ. of Michigan, Ann Arbor, Mich., 1958, 456 pp. | MR

[27] Lerman L. M., Umanskiǐ Ya. L., “Structure of the Poisson action of $\mathbf{R}^2$ on a four-dimensional symplectic manifold: 1”, Selecta Math. Sov., 6:4 (1987), 365–396 | MR | Zbl

[28] Bolsinov A. V., Fomenko A. T., Integrable Hamiltonian systems: Geometry, topology, classification, Chapman Hall, Boca Raton, Fla., 2004, 752 pp. | MR | MR | Zbl

[29] Bolsinov A. V., Borisov A. V., Mamaev I. S., “Topology and stability of integrable systems”, Russian Math. Surveys, 65:2 (2010), 259–318 | DOI | DOI | MR | Zbl