Geodesic
One family of conformally Hamiltonian systems
Teoretičeskaâ i matematičeskaâ fizika, Tome 173 (2012) no. 2, pp. 179-196
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We propose a method for constructing conformally Hamiltonian systems of dynamical equations whose invariant measure arises from the Hamiltonian equations of motion after a change of variables including a change of time. As an example, we consider the Chaplygin problem of the rolling ball and the Veselova system on the Lie algebra $e^*(3)$ and prove their complete equivalence.
Keywords: integrable system, nonholonomic system, Chaplygin ball, Veselova system. 
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A. V. Tsiganov. One family of conformally Hamiltonian systems. Teoretičeskaâ i matematičeskaâ fizika, Tome 173 (2012) no. 2, pp. 179-196. http://geodesic.mathdoc.fr/item/TMF_2012_173_2_a0/

[1] S. A. Chaplygin, “O katanii shara po gorizontalnoi ploskosti.”, Sobranie sochinenii, v. 1, Teoreticheskaya mekhanika. Matematika, OGIZ, M.–L., 1948, 76–101 | MR | Zbl

[2] A. V. Borisov, I. S. Mamaev, Regul. Chaotic Dyn., 13:5 (2008), 443–490 | DOI | MR | Zbl

[3] K. Ehlers, J. Koiller, R. Montgomery, P. Rios, “Nonholonomic systems via moving frames: Cartan equivalence and Chaplygin Hamiltonization”, The Breadth of Symplectic and Poisson Geometry. Festschrift in Honor of Alain Weinstein, Progress in Mathematics, 232, eds. J. E. Marsden, T. S. Ratiu, Birkhäuser, Boston, 2005, 75–120 | DOI | MR | Zbl

[4] Yu. N. Fedorov, B. Jovanović, Regul. Chaotic Dyn., 14:4–5 (2009), 495–505 | DOI | MR | Zbl

[5] S. Hochgerner, Differ. Geom. Appl., 28:4 (2010), 436–453 | DOI | MR | Zbl

[6] B. Jovanović, J. Nonlin. Sci., 20:5 (2010), 569–593 | DOI | MR | Zbl

[7] M. de León, J. C. Marrero, D. M. de Diego, J. Geom. Mech., 2:2 (2010), 159–198 | DOI | MR | Zbl

[8] T. Ohsawa, O. E. Fernandez, A. M. Bloch, D. V. Zenkov, J. Geom. Phys., 61:8 (2011), 1263–1291, arXiv: 1102.4361 | DOI | MR | Zbl

[9] L. E. Veselova, “Novye sluchai integriruemosti uravnenii dvizheniya tverdogo tela pri nalichii negolonomnoi svyazi”, Geometriya, differentsialnye uravneniya i mekhanika, Izd-vo MGU, M., 1986, 64–68 | MR

[10] A. P. Veselov, L. E. Veselova, Matem. zametki, 44:5 (1988), 604–619 | DOI | MR | Zbl

[11] A. P. Veselov, Vestn. Mosk. un-ta. Ser. 1. Matem., mekh., 42:5 (1987), 25–29 | MR | Zbl

[12] C.-M. Marle, A property of conformally Hamiltonian vector fields; application to the Kepler problem, 2010, arXiv: 1011.5731 | MR

[13] A. V. Borisov, I. S. Mamaev, Matem. zametki, 70:5 (2001), 793–795 | DOI | MR | Zbl

[14] A. V. Borisov, A. A. Kilin, I. S. Mamaev, Nelineinaya dinam., 7:2 (2011), 313–338 | MR

[15] Yu. N. Fedorov, Vestn. MGU. Cer. 1. Matematika. Mekhanika, 44:4 (1989), 38–41 | MR | Zbl

[16] A. V. Tsiganov, Regul. Chaotic Dyn., 17:1 (2012), 72–96, arXiv: 1106.1952 | DOI | MR | Zbl

[17] A. G. Reiman, M. A. Semenov-Tyan-Shanskii, Integriruemye sistemy. Teoretiko- gruppovoi podkhod, RKhD, Moskva–Izhevsk, 2003

[18] A. V. Bolsinov, B. Jovanović, Math. Z., 246:1–2 (2004), 213–236 | DOI | MR | Zbl

[19] A. V. Bolsinov, “Polnye involyutivnye nabory polinomov v puassonovykh algebrakh: dokazatelstvo gipotezy Mischenko–Fomenko”, Tr. sem. po vektornomu i tenzornomu analizu, 26, Izd-vo MGU, M., 2005, 87–109

[20] F. J. Turiel, C. R. Acad. Sci. Paris Sér. I, 301:20 (1985), 923–925 | MR | Zbl

[21] A. V. Bolsinov, B. Jovanović, Comment. Math. Helv., 83:3 (2008), 679–700 | DOI | MR | Zbl

[22] A. V. Tsiganov, J. Math. Phys., 40:1 (1999), 279–298 | DOI | MR | Zbl

[23] O. I. Bogoyavlenskii, Izv. AN SSSR. Ser. matem., 49:5 (1985), 899–915 | DOI | MR | Zbl

[24] S. Wojciechowski, Phys. Lett. A, 107:3 (1985), 106–111 | DOI | MR | Zbl

[25] A. V. Tsiganov, J. Geom. Mech., 3:3 (2011), 337–362 | DOI | MR | Zbl

[26] V. G. Marikhin, V. V. Sokolov, Nelineinaya dinam., 4:3 (2008), 313–332