Geodesic
The Toda chain as a reduced system
Teoretičeskaâ i matematičeskaâ fizika, Tome 45 (1980) no. 1, pp. 3-18 Cet article a éte moissonné depuis la source Math-Net.Ru

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It is proved that the phase space of a finite Toda chain can be obtained by means of the reduction of the phase space of a geodesic flow on the uniform space of symmetric positive-definite matrices. The reduction is performed with the aid of the group of triangular matrices and makes it possible to obtain explicit formulas for solutions of the equations of motion. The construction is extended to the generalised Toda chains corresponding to an arbitrary system of roots. 
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M. A. Olshanetsky; A. M. Perelomov. The Toda chain as a reduced system. Teoretičeskaâ i matematičeskaâ fizika, Tome 45 (1980) no. 1, pp. 3-18. http://geodesic.mathdoc.fr/item/TMF_1980_45_1_a0/

[1] J. M. Souriau, Structure des systems dynamiques, Dynod, Paris, 1970 | MR

[2] V. I. Arnold, Matematicheskie metody klassicheskoi mekhaniki, «Nauka», M., 1974 | MR

[3] J. Marsden, A. Weinstein, Rep. Math. Phys., 5 (1974), 121 | DOI | MR | Zbl

[4] J. Moser, Preprint of Courant Institute of Mathematical Science, New York University, 1978 | MR

[5] D. Kazhdan, B. Kostant, S. Sternberg, Commun. Pure. Appl. Math., 31 (1978), 481 | DOI | MR | Zbl

[6] M. A. Olshanetskii, A. M. Perelomov, Funkts. analiz i ego prilozh., 10 (1976), 86

[7] M. A. Olshanetskii, A. M. Perelomov, Funkts. analiz i ego prilozh., 11 (1977), 75

[8] M. A. Olshanetsky, V.-B. K. Rogov, Ann. Inst. H. Poincarè, 29 (1978), 169 | MR | Zbl

[9] M. Toda, J. Phys. Soc. Japan, 22 (1967), 431 ; 23 (1968), 501 | DOI | DOI

[10] M. Toda, Progr. Theor. Phys. Suppl., 45 (1970), 174 | DOI

[11] M. Henon, Phys. Rev., B9 (1974), 1921 | DOI | MR | Zbl

[12] S. V. Manakov, ZhETF, 67 (1974), 543 | MR

[13] H. Flaschka, Phys. Rev., B9 (1974), 1924 | DOI | MR | Zbl

[14] H. Flaschka, Progr. Theor. Phys., 51 (1974), 703 | DOI | MR | Zbl

[15] J. Moser, Lect. Notes Phys., 38 (1975), 97 | DOI

[16] B. A. Dubrovin, V. B. Matveev, S. P. Novikov, UMN, 31:1 (1976), 55 | MR | Zbl

[17] T. Kotera, S. Yamazaki, Toda lattice and Kac-Moerbeke's equation; the general solutions for the scattering problems, Preprint Inst of Phys., Univ. of Tsukuba, Niihari-gan, Ibaraki, 1976 | MR

[18] M. Adler, Invent. Math., 50 (1979), 219 | DOI | MR | Zbl

[19] A. G. Reiman, M. A. Semenov-Tyan-Shanskii, I. E. Frenkel, DAN SSSR, 244 (1979), 55 | MR

[20] I. M. Krichever, UMN, 33 (1978), 215 | MR | Zbl

[21] M. A. Olshanetsky, A. M. Perelomov, Preprint ITEP-157, Moscow, 1978 | Zbl

[22] I. M. Gelfand, M. A. Naimark, Unitarnye predstavleniya klassicheskikh grupp, Tr. MIAN SSSR, 36, 1950 | MR | Zbl

[23] A. A. Kirillov, Elementy teorii predstavlenii, «Nauka», 1972 | MR

[24] O. I. Bogoyavlensky, Commun. Math. Phys., 51 (1976), 201 | DOI | MR

[25] S. Khelgason, Differentsialnaya geometriya i simmetricheskie prostranstva, «Mir», 1964 | Zbl

[26] M. A. Olshanetskii, A. M. Perelomov, Preprint ITEF-159, Moskva, 1977

[27] B. Kostant, Preprint MIT, 1979