Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 21 (2025), 066, 19 pages           https://doi.org/10.3842/SIGMA.2025.066

Symmetric Separation of Variables for the Extended Clebsch and Manakov Models

Taras Skrypnyk
Bogolyubov Institute for Theoretical Physics, 14-b Metrolohichna Str., Kyiv, 03680, Ukraine

Received September 09, 2024, in final form July 30, 2025; Published online August 05, 2025

Abstract
In the present paper, using a modification of the method of vector fields $Z_i$ of the bi-Hamiltonian theory of separation of variables (SoV), we construct symmetric non-Stäckel variable separation for three-dimensional extension of the Clebsch model, which is equivalent (in the bi-Hamiltonian sense) to the system of interacting Manakov (Schottky-Frahm) and Euler tops. For the obtained symmetric SoV (contrary to the previously constructed asymmetric one), all curves of separation are the same and have genus five. It occurred that the difference between the symmetric and asymmetric cases is encoded in the different form of the vector fields $Z$ used to construct separating polynomial. We explicitly construct coordinates and momenta of separation and Abel-type equations in the considered examples of symmetric SoV for the extended Clebsch and Manakov models.

Key words: integrable system; separation of variables; anisotropic top.

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References

  1. Agostinelli C., Sopra l'integrazione per separazione di variabili dell'equazione dinamica di Hamilton-Jacobi, Memorie Acc. Scienze Torino 15 (1937), 3-54.
  2. Benenti S., Intrinsic characterization of the variable separation in the Hamilton-Jacobi equation, J. Math. Phys. 38 (1997), 6578-6602.
  3. Benenti S., Chanu C., Rastelli G., Variable separation for natural Hamiltonians with scalar and vector potentials on Riemannian manifolds, J. Math. Phys. 42 (2001), 2065-2091.
  4. Benenti S., Chanu C., Rastelli G., Remarks on the connection of between the additive separation of the Hamilton-Jacobi equation and the multiplicative separation of the Schroedinger equation. II. First integrals and symmetry operators, J. Math. Phys. 43 (2002), 5183-5222.
  5. Blaszak M., Marciniak K., Algebraic curves as a source of separable multi-Hamiltonian systems, Open Commun. Nonlinear Math. Phys. 2 (2024), 12861, 27 pages, arXiv:2401.04673.
  6. Clebsch A., Ueber die Bewegung eines Körpers in einer Flüssigkeit, Math. Ann. 3 (1870), 238-262.
  7. Dubrovin B., Skrypnyk T., Separation of variables for linear Lax algebras and classical $r$-matrices, J. Math. Phys. 59 (2018), 091405, 39 pages.
  8. Falqui G., Magri F., Pedroni M., Bihamiltonian geometry and separation of variables for Toda lattices, J. Nonlinear Math. Phys. 8 (2001), suppl. 1, 118-127, arXiv:nlin.SI/0002008.
  9. Falqui G., Pedroni M., On a Poisson reduction for Gelfand-Zakharevich manifolds, Rep. Math. Phys. 50 (2002), 395-407, arXiv:nlin.SI/0204050.
  10. Falqui G., Pedroni M., Separation of variables for bi-Hamiltonian systems, Math. Phys. Anal. Geom. 6 (2003), 139-179, arXiv:nlin.SI/0204029.
  11. Fedorov Yu., Magri F., Skrypnyk T., A new approach to separation of variables for the Clebsch integrable system. Part II: Inversion of the Abel-Prym map, arXiv:2102.03599.
  12. Gelfand I.M., Zakharevich I., Webs, Lenard schemes, and the local geometry of bi-Hamiltonian Toda and Lax structures, Selecta Math. (N.S.) 6 (2000), 131-183, arXiv:math.DG/9903080.
  13. Levi-Civita T., Sulla integrazione della equazione di Hamilton-Jacobi per separazione di variabili, Math. Ann. 59 (1904), 383-397.
  14. Magri F., Skrypnyk T., The Clebsch system, arXiv:1512.04872.
  15. Sklyanin E.K., Separation of variables: New trends, Progr. Theoret. Phys. Suppl. 118 (1995), 35-60, arXiv:solv-int/9504001.
  16. Skrypnyk T., ''Symmetric'' separation of variables for the Clebsch system, J. Geom. Phys. 135 (2019), 204-218.
  17. Skrypnyk T., On a class of $gl(n)\otimes gl(n)$-valued classical $r$-matrices and separation of variables, J. Math. Phys. 62 (2021), 063508, 27 pages.
  18. Skrypnyk T., Symmetric and asymmetric separation of variables for an integrable case of the complex Kirchhoff's problem, J. Geom. Phys. 172 (2022), 104418, 26 pages.
  19. Skrypnyk T., Symmetric and asymmetric separated variables. II. A second integrable case of the complex Kirchhoff model, J. Math. Phys. 64 (2023), 093503, 21 pages.
  20. Skrypnyk T., Asymmetric separation of variables for the extended Clebsch and Manakov models, J. Geom. Phys. 197 (2024), 105078, 18 pages.
  21. Skrypnyk T., Separation of variables for the Clebsch model: $so(4)$ spectral/separation curves, J. Geom. Phys. 212 (2025), 105453, 11 pages.
  22. Stäckel P., Ueber die Bewegung eines Punktes in einer $n$-fachen Mannigfaltigkeit, Math. Ann. 42 (1893), 537-563.
  23. Stäckel P., Ueber quadratische Integrale der Differentialgleichungen der Dynamik, Ann. Mat. Pura Appl. 26 (1897), 55-60.

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