Geodesic
Canonical transformations of the extended phase space and integrable systems
Teoretičeskaâ i matematičeskaâ fizika, Tome 124 (2000) no. 1, pp. 72-94 Cet article a éte moissonné depuis la source Math-Net.Ru

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We investigate the explicit construction of a canonical transformation of the time variable and the Hamiltonian whereby a given completely integrable system is mapped into another integrable system. The change of time induces a transformation of the equations of motion and of their solutions, the integrals of motion, the methods of separation of variables, the Lax matrices, and the corresponding $r$-matrices. For several specific families of integrable systems (Toda chains, Holt systems, and Stäckel-type systems), we construct canonical transformations of time in the extended phase space that preserve the integrability property. 
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A. V. Tsiganov. Canonical transformations of the extended phase space and integrable systems. Teoretičeskaâ i matematičeskaâ fizika, Tome 124 (2000) no. 1, pp. 72-94. http://geodesic.mathdoc.fr/item/TMF_2000_124_1_a5/

[1] A. G. Reiman, M. A. Semenov-Tyan-Shanskii, “Teoretiko-gruppovye metody v teorii integriruemykh sistem”, Dinamicheskie sistemy–7, Itogi nauki i tekhniki. Sovremennye problemy matematiki. Fundamentalnye napravleniya, 16, ed. R. V. Gamkrelidze, VINITI, M., 1987, 119 | MR

[2] P. Stäckel, Compt. Rend. Acad. Sci. (Paris), 116 (1893), 485; 1284 | Zbl

[3] K. Lantsosh, Variatsionnye printsipy mekhaniki, Mir, M., 1965

[4] Dzh. L. Sing, Klassicheskaya dinamika, Fizmatgiz, M., 1963

[5] A. M. Perelomov, Integriruemye sistemy klassicheskoi mekhaniki i algebry Li, Nauka, M., 1990 | Zbl

[6] T. Levi-Civita, Acta Math., 30 (1906), 305 | DOI | MR | Zbl

[7] J. Moser, Commun. Pure Appl. Math., 23 (1970), 609 | DOI | MR | Zbl

[8] A. V. Tsyganov, TMF, 120 (1999), 27 | DOI | MR | Zbl

[9] V. A. Fock, Z. Physik, 98 (1935), 145 | DOI | Zbl

[10] E. Schrödinger, Proc. Roy. Irish. Acad. A, 46 (1940), 9 ; 183 ; 47 (1941), 53 | Zbl | Zbl | Zbl

[11] M. Rid, B. Saimon, Metody sovremennoi matematicheskoi fiziki, T. 3, Mir, M., 1982 | MR

[12] A. V. Tsyganov, TMF, 118 (1999), 205 | DOI | MR | Zbl

[13] L. D. Faddeev, L. A. Takhtadzhyan, Gamiltonov podkhod v teorii solitonov, Nauka, M., 1986 | MR

[14] H. Flaschka, D. McLaughlin, Bäcklund Transformations, Lect. Notes Math., 515, ed. R. Miura, Springer, Berlin, 1976, 225 | MR

[15] E. K. Sklyanin, V. B. Kuznetsov, J. Phys. A, 31 (1998), 2241 | DOI | MR | Zbl

[16] V. B. Kuznetsov, A. V. Tsyganov, Zap. nauchn. semin. LOMI, 172, 1989, 88

[17] J. Drach, Compt. Rend. Acad. Sci. (Paris), 200 (1935), 22 | Zbl

[18] C. R. Holt, J. Math. Phys., 23 (1982), 37 | DOI | MR

[19] A. Ramani, B. Grammaticos, T. Bountis, Phys. Rep., 180 (1989), 159 | DOI | MR

[20] M. Blaszak, S. Rauch-Wojciechowski, J. Math. Phys., 35 (1994), 1693 | DOI | MR | Zbl

[21] K. Yakobi, Lektsii po dinamike, Glav. red. obschetekhnicheskoi literatury, M.–L., 1936

[22] A. V. Tsyganov, TMF, 115 (1998), 3 | DOI | MR | Zbl

[23] J. Hietarinta, B. Grammaticos, B. Dorizzi, A. Ramani, Phys. Rev. Lett., 53 (1984), 1707 | DOI | MR

[24] L. Pars, Am. Math. Monthly, 56 (1949), 395 | DOI | MR

[25] A. S. Fokas, P. Lagerstrom, J. Math. Anal. Appl., 74 (1980), 325 | DOI | MR | Zbl