Geodesic
Hamiltonisation of non-holonomic systems in the neighborhood of invariant manifolds
Russian journal of nonlinear dynamics, Tome 6 (2010) no. 4, pp. 829-854.

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Hamiltonisation problem for non-holonomic systems, both integrable and non-integrable, is considered. This question is important for qualitative analysis of such systems and allows one to determine possible dynamical effects. The first part is devoted to the representation of integrable systems in a conformally Hamiltonian form. In the second part, the existence of a conformally Hamiltonian representation in a neighbourhood of a periodic solution is proved for an arbitrary measure preserving system (including integrable). General consructions are always illustrated by examples from non-holonomic mechanics.
Keywords: conformally Hamiltonian system, nonholonomic system, invariant measure, periodic trajectory, integrable system.
Mots-clés : invariant torus 
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A. V. Bolsinov; A. V. Borisov; I. S. Mamaev. Hamiltonisation of non-holonomic systems in the neighborhood of invariant manifolds. Russian journal of nonlinear dynamics, Tome 6 (2010) no. 4, pp. 829-854. http://geodesic.mathdoc.fr/item/ND_2010_6_4_a7/

[30] Arnold V. I., Matematicheskie metody klassicheskoi mekhaniki, 3-e izd., Nauka, M., 1989, 472 pp.

[31] Arnold V. I., Kozlov V. V., Neishtadt A. I., Matematicheskie aspekty klassicheskoi i nebesnoi mekhaniki, URSS, M., 2002, 414 pp.

[32] Bolsinov A. V., “Gladkaya traektornaya klassifikatsiya integriruemykh gamiltonovykh sistem s dvumya stepenyami svobody”, Matem. sb., 186:1 (1995), 3–28 | MR

[33] Bolsinov A. V., Fomenko A. T., Integriruemye gamiltonovy sistemy. Geometriya, topologiya, klassifikatsiya, v 2-x tt., RKhD, M.–Izhevsk, 1999, 444 pp.; Bolsinov A. V., Fomenko A. T., Integrable Hamiltonian systems: Geometry, topology, classification, Chapman Hall, Boca Raton, FL, 2004, 730 pp. | MR | Zbl

[34] Bolsinov A. V., Fomenko A. T., “Traektornaya ekvivalentnost integriruemykh gamiltonovykh sistem s dvumya stepenyami svobody. Teorema klassifikatsii: I”, Matem. sb., 185:4 (1994), 27–80

[35] Borisov A. V., Mamaev I. S., “Gamiltonovost zadachi Chaplygina o kachenii shara”, Matem. zametki, 70:5 (2001), 793–795 | MR | Zbl

[36] Borisov A. V., Mamaev I. S., “O nesuschestvovanii invariantnoi mery pri kachenii neodnorodnogo ellipsoida po ploskosti”, Matem. zametki, 77:6 (2005), 930–932 | MR | Zbl

[37] Borisov A. V., Mamaev I. S., “Prepyatstvie k gamiltonovosti negolonomnykh sistem”, Dokl. RAN, 387:6 (2002), 764–766

[38] Borisov A. V., Mamaev I. S., Marikhin V. G., “Yavnoe integrirovanie odnoi negolonomnoi zadachi”, Dokl. RAN, 422:4 (2008), 475–478 | MR | Zbl

[39] Borisov A. V., Fëdorov Yu. N., “O dvukh vidoizmenennykh integriruemykh zadachakh dinamiki”, Vestn. Mosk. un-ta. Ser. 1. Matem. Mekhan., 1995, no. 6, 102–105 | MR

[40] Veselova L. E., “Novye sluchai integriruemosti uravnenii dvizheniya tverdogo tela pri nalichii negolonomnoi svyazi”, Geometriya, differentsialnye uravneniya i mekhanika, Sb. st., eds. V. V. Kozlov, A. T. Fomenko, MGU, M., 1986, 64–68

[41] Kozlov V. V., “Dve integriruemye zadachi klassicheskoi dinamiki”, Vestn. Mosk. un-ta. Ser. 1. Matem. Mekhan., 1981, no. 4, 80–83 | MR

[42] Kozlov V. V., “K teorii integrirovaniya uravnenii negolonomnoi mekhaniki”, Uspekhi mekhan., 8:3 (1985), 85–107 ; Неголономные динамические системы, под ред. А. В. Борисова, И. С. Мамаева, Инст. компьютерн. исслед., М.–Ижевск, 2002 | MR | MR

[43] Kozlov V. V., “O suschestvovanii integralnogo invarianta gladkikh dinamicheskikh sistem”, PMM, 51:4 (1987), 538–545 | MR | Zbl

[44] Kolmogorov A. N., “O dinamicheskikh sistemakh s integralnym invariantom na tore”, Dokl. AN SSSR, 93:5 (1953), 763–766 | MR | Zbl

[45] Novikov S. P., “Gamiltonov formalizm i mnogoznachnyi analog teorii Morsa”, UMN, 37:5(227) (1982), 3–49 | MR

[46] Sinai Ya. G., Vvedenie v ergodicheskuyu teoriyu, 2-e izd., Fazis, M., 1996, 128 pp.

[47] Fomenko A. T., Tsishang Kh., “Topologicheskii invariant i kriterii ekvivalentnosti integriruemykh gamiltonovykh sistem s dvumya stepenyami svobody”, Izv. AN SSSR. Ser. Matem., 54:3 (1990), 546–575 | Zbl

[48] Chaplygin S. A., “O katanii shara po gorizontalnoi ploskosti”, Matem. sb., 24:1 (1903), 139–168 | Zbl

[49] Shubin M. A., Psevdodifferentsialnye operatory i spektralnaya teoriya, 2-e izd., Dobrosvet, M., 2005, 312 pp.

[50] Berndt R., An introduction to symplectic geometry, Grad. Stud. Math., 26, AMS, Providence, RI, 2001, 195 pp. | MR | Zbl

[51] Borisov A. V., Fedorov Yu. N., Mamaev I. S., “Chaplygin ball over a fixed sphere: An explicit integration”, Regul. Chaotic Dyn., 13:6 (2008), 557–571 | DOI | MR

[52] Borisov A. V., Kilin A. A., Mamaev I. S., “Stability of steady rotations in the nonholonomic Routh problem”, Regul. Chaotic Dyn., 13:4 (2008), 239–249 | DOI | MR

[53] Borisov A. V., Mamaev I. S., “Conservation laws, hierarchy of dynamics and explicit integration of nonholonomic systems”, Regul. Chaotic Dyn., 13:5 (2008), 443–490 | DOI | MR

[54] Borisov A. V., Mamaev I. S., “Isomorphism and Hamilton representation of some nonholonomis systems”, Siberian Math. J., 48:1 (2007), 26–36 ; (Sept. 21, 2005), arXiv: nlin.-SI/0509036 v. 1 | DOI | MR | Zbl

[55] Borisov A. V., Mamaev I. S., “Rolling of a non-homogeneous ball over a sphere without slipping and twisting”, Regul. Chaotic Dyn., 12:2 (2007), 153–159 | DOI | MR

[56] Borisov A. V., Mamaev I. S., “The rolling motion of a rigid body on a plane and sphere: Hierarchy of dynamics”, Regul. Chaotic Dyn., 7:2 (2002), 177–200 | DOI | MR | Zbl

[57] Borisov A. V., Mamaev I. S., Kilin A. A., “The rolling motion of a ball on a surface: New integrals and hierarchy of dynamics”, Regul. Chaotic Dyn., 7:2 (2002), 201–219 | DOI | MR | Zbl

[58] Cushman R., “Routh's sphere”, Rep. Math. Phys., 42:1 (1998), 47–70 | DOI | MR | Zbl

[59] Cushman R., Duistermaat J. J., “Non-Hamiltonian monodromy”, J. Differential Equations, 172:1 (2001), 42–58 | DOI | MR | Zbl

[60] Ehlers K., Koiller J., “Rubber rolling: Geometry and dynamics of $2-3-5$ distributions”, In Proceedings IUTAM symposium 2006 on Hamiltonian Dynamics, Vortex Structures, Turbulence (Steklov Institute, Moscow) (to appear)

[61] Ehlers K., Koiller J., Montgomery R., Rios P., “Nonholonomic systems via moving frames: Cartan equivalence and Chaplygin Hamiltonization”, The breadth of symplectic and Poisson geometry: Festschrift in honor of Alain Weinstein, Progr. Math., 232, eds. J. E. Marsden, T. S. Ratiu, Birkhäuser, Boston, 2005, 75–120 | MR | Zbl

[62] Fedorov Yu. N., Jovanović B., “Nonholomonic $LR$-systems as generalized Chaplygin systems with an invariant measure and flows on homogeneous spaces”, J. Nonlinear Sci., 1:14 (2004), 341–381 | MR

[63] Koiller J., Ehlers K., “Rubber rolling over a sphere”, Regul. Chaotic Dyn., 12:2 (2007), 127–152 | DOI | MR

[64] McDuff D., Salamon D., Introduction to symplectis topology, Oxford Univ. Press, Oxford, 1998, 512 pp. | MR | Zbl

[65] Pronin A. V., Treschev D. V., “On the inclusion of analytic maps into analytic flows”, Regul. Chaotic Dyn., 2:2 (1997), 14–24 | MR | Zbl

[66] Siegel C. L., “Note on differential equations on the torus”, Ann. of Math. (2), 46:3 (1945), 423–428 | DOI | MR | Zbl