Abstract
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.
Similar content being viewed by others
References
F. Frahm, “Ueber gewisse Differentialgleichungen,” Math. Ann., 8, No. 1, 35–44 (1874).
F. Shottky, “Uber das analitische Problem der Rotation eines starren Körpers in Raume von vier Dimensionen,” Sitzunber. Königl. Preuss. Akad. Wiss., 13, 227–232 (1891).
S. Manakov, “Note on the integration of Euler’s equations of the dynamics of an n-dimensional rigid body,” Funct. Anal. Appl., 10, 328–329 (1976).
A. Perelomov, “Motion of four-dimensional rigid body around a fixed point: An elementary approach I,” J. Phys. A: Math. Gen., 38, L801–L807 (2005).
A. Clebsch, “Ueber die Bewegung eines Körpers in einer Flüssigkeit,” Math. Ann., 3, 238–261 (1871).
F. Kötter, “Ueber die Bewegung eines festen Körpers in einer Flüssigkeit,” J. Reine Angew. Math., 1892, No. 109, 51–81, 89–111 (2009).
E. Sklyanin and T. Takebe, “Separation of variables in the elliptic Gaudin model,” Commun. Math. Phys., 204, 17–38 (1999).
M. Gaudin, “Diagonalisation d’une classe d’Hamiltoniens de spin,” J. Physique, 37, 1087–1098 (1976).
V. G. Marikhin and V. V. Sokolov, “Separation of variables on a non-hyperelliptic curve,” Reg. Chaot. Dyn., 10, 59–75 (2005).
F. Magri and T. Skrypnyk, “The Clebsch system,” arXiv:1512.04872v1 [nlin.SI] (2015).
F. Magri, “A simple model of the integrable Hamiltonian equation,” J. Math. Phys., 19, 1156 (1978).
G. Falqui, F. Magri, and M. Pedroni, “Bihamiltonian geometry and separation of variables for Toda lattices,” J. Nonlinear Math. Phys., 8, suppl., 118–127 (2001).
G. Falqui and M. Pedroni, “On a Poisson reduction for Gel’fand–Zakharevich manifolds,” Rep. Math. Phys., 50, 395–407 (2002).
L. Haine, “Geodesic flow on SO(4) and abelian surfaces,” Math. Ann., 263, 435–472 (1983).
V. Z. Enolsky and Yu. N. Fedorov, “Algebraic description of Jacobians isogeneous to certain Prym varieties with polarization (1, 2),” Exp. Math., 27, 147–178 (2018).
Y. Fedorov, private communications, article in preparation.
A. Tsiganov, “On the invariant separated variables,” Reg. Chaot. Dyn., 6, 307–326 (2001).
G. Falqui, “A note on the rotationally symmetric SO(4) Euler rigid body,” SIGMA, 3, 032 (2007).
E. Sklyanin, “Separation of variables: New trends,” Progr. Theor. Phys. Suppl., 118, 35–60 (1995).
Author information
Authors and Affiliations
Corresponding author
Additional information
Prepared from an English manuscript submitted by the author; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, Vol. 196, No. 3, pp. 465–486, September, 2018.
Rights and permissions
About this article
Cite this article
Skrypnyk, T.V. Separation of Variables in the Anisotropic Shottky–Frahm Model. Theor Math Phys 196, 1347–1365 (2018). https://doi.org/10.1134/S0040577918090088
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0040577918090088