Geodesic
Pairs of commuting Hamiltonians quadratic in the momenta
Teoretičeskaâ i matematičeskaâ fizika, Tome 149 (2006) no. 2, pp. 147-160

Voir la notice de l'article provenant de la source Math-Net.Ru

In the case of two degrees of freedom, we consider pairs of Hamiltonians quadratic in the momenta and commuting with respect to the standard Poisson bracket. We find new multiparameter families of such pairs and present a universal scheme for constructing a complete solution of the Hamilton–Jacobi equation in terms of integrals over an algebraic curve. For the most complicated examples, this curve is a nonhyperelliptic covering of an elliptic curve.
Keywords: integrable Hamiltonian system, separation of variables, algebraic system. 
V. G. Marikhin; V. V. Sokolov. Pairs of commuting Hamiltonians quadratic in the momenta. Teoretičeskaâ i matematičeskaâ fizika, Tome 149 (2006) no. 2, pp. 147-160. http://geodesic.mathdoc.fr/item/TMF_2006_149_2_a0/
@article{TMF_2006_149_2_a0,
     author = {V. G. Marikhin and V. V. Sokolov},
     title = {Pairs of commuting {Hamiltonians} quadratic in the~momenta},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {147--160},
     year = {2006},
     volume = {149},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2006_149_2_a0/}
}
TY  - JOUR
AU  - V. G. Marikhin
AU  - V. V. Sokolov
TI  - Pairs of commuting Hamiltonians quadratic in the momenta
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2006
SP  - 147
EP  - 160
VL  - 149
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/TMF_2006_149_2_a0/
LA  - ru
ID  - TMF_2006_149_2_a0
ER  - 
%0 Journal Article
%A V. G. Marikhin
%A V. V. Sokolov
%T Pairs of commuting Hamiltonians quadratic in the momenta
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2006
%P 147-160
%V 149
%N 2
%U http://geodesic.mathdoc.fr/item/TMF_2006_149_2_a0/
%G ru
%F TMF_2006_149_2_a0

[1] B. Dorizzi, B. Grammaticos, A. Ramani, P. Winternitz, J. Math. Phys., 26 (1985), 3070–3079 ; E. V. Ferapontov, A. P. Fordy, Physica D, 108 (1997), 350–364 ; Rep. Math. Phys., 44:1–2 (1999), 71–80 ; E. McSween, P. Winternitz, J. Math. Phys., 41 (2000), 2957–2967 | DOI | MR | DOI | MR | Zbl | DOI | MR | Zbl | DOI | MR | Zbl

[2] H. M. Yehia, J. Phys. A, 25 (1992), 197–221 | DOI | MR | Zbl

[3] V. G. Marikhin, V. V. Sokolov, Regul. Chaotic Dyn., 10:1 (2005), 59–70 | DOI | MR | Zbl

[4] V. G. Marikhin, V. V. Sokolov, UMN, 60:5 (2005), 175–176 | DOI | MR | Zbl

[5] F. Schottky, Sitzungsber. König. Preuss. Acad. Wiss. zu Berlin, 13 (1891), 227–232; С. В. Манаков, Функц. анализ и его прилож., 10:4 (1976), 93–94 ; A. Clebsch, Math. Ann., 3 (1870), 238–262 | MR | Zbl | DOI | MR

[6] V. V. Prasolov, Yu. P. Solovev, Ellipticheskie funktsii i algebraicheskie uravneniya, Faktorial, M., 1997

[7] V. A. Stekloff, Ann. Fac. Sci. Toulouse Sci. Math. Sci. Phys., Ser. 3, 1 (1909), 145–226 | MR