Geodesic
Non-Homogeneous Hydrodynamic Systems and Quasi-Stäckel Hamiltonians
Symmetry, integrability and geometry: methods and applications, Tome 13 (2017) Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

In this paper we present a novel construction of non-homogeneous hydrodynamic equations from what we call quasi-Stäckel systems, that is non-commutatively integrable systems constructed from appropriate maximally superintegrable Stäckel systems. We describe the relations between Poisson algebras generated by quasi-Stäckel Hamiltonians and the corresponding Lie algebras of vector fields of non-homogeneous hydrodynamic systems. We also apply Stäckel transform to obtain new non-homogeneous equations of considered type.
Keywords: Hamiltonian systems; superintegrable systems; Stäckel systems; hydrodynamic systems; Stäckel transform. 
@article{SIGMA_2017_13_a76,
     author = {Krzysztof Marciniak and Maciej B{\l}aszak},
     title = {Non-Homogeneous {Hydrodynamic} {Systems} and {Quasi-St\"ackel} {Hamiltonians}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2017},
     volume = {13},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2017_13_a76/}
}
TY  - JOUR
AU  - Krzysztof Marciniak
AU  - Maciej Błaszak
TI  - Non-Homogeneous Hydrodynamic Systems and Quasi-Stäckel Hamiltonians
JO  - Symmetry, integrability and geometry: methods and applications
PY  - 2017
VL  - 13
UR  - http://geodesic.mathdoc.fr/item/SIGMA_2017_13_a76/
LA  - en
ID  - SIGMA_2017_13_a76
ER  - 
%0 Journal Article
%A Krzysztof Marciniak
%A Maciej Błaszak
%T Non-Homogeneous Hydrodynamic Systems and Quasi-Stäckel Hamiltonians
%J Symmetry, integrability and geometry: methods and applications
%D 2017
%V 13
%U http://geodesic.mathdoc.fr/item/SIGMA_2017_13_a76/
%G en
%F SIGMA_2017_13_a76
Krzysztof Marciniak; Maciej Błaszak. Non-Homogeneous Hydrodynamic Systems and Quasi-Stäckel Hamiltonians. Symmetry, integrability and geometry: methods and applications, Tome 13 (2017). http://geodesic.mathdoc.fr/item/SIGMA_2017_13_a76/

[1] Błaszak M., “Separable systems with quadratic in momenta first integrals”, J. Phys. A: Math. Gen., 38 (2005), 1667–1685, arXiv: nlin.SI/0312025 | DOI | MR

[2] Błaszak M., Marciniak K., “From {S}täckel systems to integrable hierarchies of PDE's: Benenti class of separation relations”, J. Math. Phys., 47 (2006), 032904, 26 pp., arXiv: nlin.SI/0511062 | DOI | MR

[3] Błaszak M., Marciniak K., “On reciprocal equivalence of Stäckel systems”, Stud. Appl. Math., 129 (2012), 26–50, arXiv: 1201.0446 | DOI | MR

[4] Błaszak M., Marciniak K., “Classical and quantum superintegrability of Stäckel systems”, SIGMA, 13 (2017), 008, 23 pp., arXiv: 1608.04546 | DOI | MR

[5] Błaszak M., Sergyeyev A., “Natural coordinates for a class of Benenti systems”, Phys. Lett. A, 365 (2007), 28–33, arXiv: nlin.SI/0604022 | DOI | MR

[6] Błaszak M., Sergyeyev A., “A coordinate-free construction of conservation laws and reciprocal transformations for a class of integrable hydrodynamic-type systems”, Rep. Math. Phys., 64 (2009), 341–354 | DOI | MR

[7] Bolsinov A. V., Jovanović B., “Noncommutative integrability, moment map and geodesic flows”, Ann. Global Anal. Geom., 23 (2003), 305–322, arXiv: math-ph/0109031 | DOI | MR | Zbl

[8] Boyer C. P., Kalnins E. G., Miller Jr. W., “Stäckel-equivalent integrable Hamiltonian systems”, SIAM J. Math. Anal., 17 (1986), 778–797 | DOI | MR | Zbl

[9] Dolan P., Kladouchou A., Card C., “On the significance of Killing tensors”, Gen. Relativity Gravitation, 21 (1989), 427–437 | DOI | MR | Zbl

[10] Ferapontov E. V., “Integration of weakly nonlinear hydrodynamic systems in Riemann invariants”, Phys. Lett. A, 158 (1991), 112–118 | DOI | MR

[11] Ferapontov E. V., Fordy A. P., “Non-homogeneous systems of hydrodynamic type, related to quadratic Hamiltonians with electromagnetic term”, Phys. D, 108 (1997), 350–364 | DOI | MR | Zbl

[12] Ferapontov E. V., Fordy A. P., “Separable Hamiltonians and integrable systems of hydrodynamic type”, J. Geom. Phys., 21 (1997), 169–182 | DOI | MR | Zbl

[13] Ferapontov E. V., Fordy A. P., “Commuting quadratic Hamiltonians with velocity dependent potentials”, Rep. Math. Phys., 44 (1999), 71–80 | DOI | MR | Zbl

[14] Hietarinta J., Grammaticos B., Dorizzi B., Ramani A., “Coupling-constant metamorphosis and duality between integrable Hamiltonian systems”, Phys. Rev. Lett., 53 (1984), 1707–1710 | DOI | MR

[15] Kalnins E. G., Miller Jr. W., Reid G. J., “Separation of variables for complex Riemannian spaces of constant curvature. I. Orthogonal separable coordinates for ${\rm S}_{nC}$ and ${\rm E}_{nC}$”, Proc. Roy. Soc. London Ser. A, 394 (1984), 183–206 | DOI | MR | Zbl

[16] Marchesiello A., Šnobl L., Winternitz P., “Three-dimensional superintegrable systems in a static electromagnetic field”, J. Phys. A: Math. Theor., 48 (2015), 395206, 24 pp., arXiv: 1507.04632 | DOI | MR | Zbl

[17] Marikhin V. G., “On three-dimensional quasi-Stäckel Hamiltonians”, J. Phys. A: Math. Theor., 47 (2014), 175201, 6 pp., arXiv: 1312.4081 | DOI | MR | Zbl

[18] Marikhin V. G., Sokolov V. V., “On quasi-Stäckel Hamiltonians”, Russian Math. Surveys, 60 (2005), 981–983 | DOI | MR | Zbl

[19] Mishchenko A. S., Fomenko A. T., “Generalized Liouville method of integration of Hamiltonian systems”, Funct. Anal. Appl., 12 (1978), 113–121 | DOI | MR | Zbl

[20] Rozhdestvenskii B. L., Sidorenko A. D., “Impossibility of the “gradient catastrophe” for slightly non-linear systems”, USSR Comput. Math. Math. Phys., 7 (1967), 282–287 | DOI | MR

[21] Sergyeyev A., Błaszak M., “Generalized Stäckel transform and reciprocal transformations for finite-dimensional integrable systems”, J. Phys. A: Math. Theor., 41 (2008), 105205, 20 pp., arXiv: 0706.1473 | DOI | MR | Zbl

[22] Tsarëv S. P., “The geometry of Hamiltonian systems of hydrodynamic type. The generalized hodograph method”, Math. USSR-Izv., 37 (1991), 397–419 | DOI | MR