Geodesic
Separation of variables in the anisotropic Shottky–Frahm model
Teoretičeskaâ i matematičeskaâ fizika, Tome 196 (2018) no. 3, pp. 465-486 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We construct separated coordinates for the completely anisotropic Shottky–Frahm model on an arbitrary coadjoint orbit of $SO(4)$. We find explicit reconstruction formulas expressing dynamical variables in terms of the separation coordinates and write the equations of motion in the Abel-type form.
Keywords: integrable system, separation of variables, classical top. 
@article{TMF_2018_196_3_a7,
     author = {T. V. Skrypnik},
     title = {Separation of variables in the~anisotropic {Shottky{\textendash}Frahm} model},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {465--486},
     year = {2018},
     volume = {196},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2018_196_3_a7/}
}
TY  - JOUR
AU  - T. V. Skrypnik
TI  - Separation of variables in the anisotropic Shottky–Frahm model
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2018
SP  - 465
EP  - 486
VL  - 196
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/TMF_2018_196_3_a7/
LA  - ru
ID  - TMF_2018_196_3_a7
ER  - 
%0 Journal Article
%A T. V. Skrypnik
%T Separation of variables in the anisotropic Shottky–Frahm model
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2018
%P 465-486
%V 196
%N 3
%U http://geodesic.mathdoc.fr/item/TMF_2018_196_3_a7/
%G ru
%F TMF_2018_196_3_a7
T. V. Skrypnik. Separation of variables in the anisotropic Shottky–Frahm model. Teoretičeskaâ i matematičeskaâ fizika, Tome 196 (2018) no. 3, pp. 465-486. http://geodesic.mathdoc.fr/item/TMF_2018_196_3_a7/

[1] F. Frahm, “Ueber gewisse Differentialgleichungen”, Math. Ann., 8:1 (1874), 35–44 | DOI

[2] F. Shottky, “Uber das analitische Problem der Rotation eines starren Körpers in Raume von vier Dimensionen”, Sitzunber. Königl. Preuss. Akad. Wiss. Berlin, 13 (1891), 227–232

[3] S. V. Manakov, “Zamechanie ob integrirovanii uravnenii Eilera dinamiki $n$-mernogo tverdogo tela”, Funkts. analiz i ego pril., 10:4 (1976), 93–94 | DOI | MR | Zbl

[4] A. Perelomov, J. Phys. A: Math. Gen., 38:47 (2005), L801–L807 | DOI | MR | Zbl

[5] A. Clebsch, “Ueber die Bewegung eines Körpers in einer Flüssigkeit”, Math. Ann., 3:2 (1871), 238–261 | DOI | MR

[6] F. Kötter, “Ueber die Bewegung eines festen Körpers in einer Flüssigkeit”, J. Reine Angew. Math., 1892:109 (2009), 51–81 ; 89–111 | DOI | MR | DOI | Zbl

[7] E. Sklyanin, T. Takebe, “Separation of variables in the elliptic Gaudin model”, Commun. Math. Phys., 204:1 (1999), 17–38 | DOI | MR | Zbl

[8] M. Gaudin, “Diagonalisation d'une classe d'Hamiltoniens de spin”, J. Physique, 37:10 (1976), 1087–1098 | DOI | MR

[9] V. G. Marikhin, V. V. Sokolov, “Separation of variables on a non-hyperelliptic curve”, Reg. Chaot. Dyn., 10:1 (2005), 59–75 | DOI | MR

[10] F. Magri, T. Skrypnyk, The Clebsch System, arXiv: 1512.04872

[11] F. Magri, “A simple model of the integrable Hamiltonian equation”, J. Math. Phys., 19 (1978), 1156 | DOI | MR | Zbl

[12] G. Falqui, F. Magri, M. Pedroni, “Bihamiltonian geometry and separation of variables for Toda lattices”, J. Nonlinear Math. Phys., 8, suppl. (2001), 118–127 | DOI | MR | Zbl

[13] G. Falqui, M. Pedroni, “On a Poisson reduction for Gel'fand–Zakharevich manifolds”, Rep. Math. Phys., 50:3 (2002), 395–407 | DOI | MR | Zbl

[14] L. Haine, “Geodesic flow on ${\rm SO}(4)$ and abelian surfaces”, Math. Ann., 263:4 (1983), 435–472 | DOI | MR | Zbl

[15] V. Z. Enolski, Yu. N. Fedorov, “Algebraic description of Jacobians isogeneous to certain Prym varieties with polarization $(1,2)$”, Exp. Math., 27:2 (2018), 147–178 | DOI

[16] Y. Fedorov, private communications, article in preparation

[17] A. Tsiganov, “On the invariant separated variables”, Reg. Chaot. Dyn., 6:3 (2001), 307–326 | DOI | MR | Zbl

[18] G. Falqui, “A note on the rotationally symmetric $\mathrm{SO}(4)$ Euler rigid body”, SIGMA, 3 (2007), 032 | Zbl

[19] E. Sklyanin, “Separation of variables: new trends”, Progr. Theor. Phys. Suppl., 118 (1995), 35–60 | DOI