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TMF, 2006, Volume 149, Number 2, Pages 147–160 (Mi tmf4224)  

This article is cited in 11 scientific papers (total in 11 papers)

Pairs of commuting Hamiltonians quadratic in the momenta

V. G. Marikhin, V. V. Sokolov

L. D. Landau Institute for Theoretical Physics, Russian Academy of Sciences

Abstract: 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

DOI: https://doi.org/10.4213/tmf4224

Full text: PDF file (417 kB)
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English version:
Theoretical and Mathematical Physics, 2006, 149:2, 1425–1436

Bibliographic databases:

Received: 24.05.2006

Citation: V. G. Marikhin, V. V. Sokolov, “Pairs of commuting Hamiltonians quadratic in the momenta”, TMF, 149:2 (2006), 147–160; Theoret. and Math. Phys., 149:2 (2006), 1425–1436

Citation in format AMSBIB
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  • http://mi.mathnet.ru/eng/tmf/v149/i2/p147

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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. Borisov AV, Mamaev IS, Marikhin VG, “Explicit Integration of One Problem in Nonholonomic Mechanics”, Doklady Physics, 53:10 (2008), 525–528  crossref  zmath  adsnasa  isi  elib  scopus
    2. Borisov AV, Mamaev IS, “Conservation Laws, Hierarchy of Dynamics and Explicit Integration of Nonholonomic Systems”, Regular & Chaotic Dynamics, 13:5 (2008), 443–490  crossref  mathscinet  zmath  adsnasa  isi  scopus
    3. I. S. Mamaev, “Universalnyi kompleks programm dlya issledovaniya mekhanicheskikh sistem s negolonomnymi svyazyami”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 2009, no. 2, 147–160  mathnet
    4. Marikhin V.G., Sokolov V.V., “Transformation of a Pair of Commuting Hamiltonians Quadratic in Momenta to Canonical Form and Real Partial Separation of Variables for the Clebsch Top”, Regular & Chaotic Dynamics, 15:6 (2010), 652–658  crossref  mathscinet  zmath  adsnasa  isi  scopus
    5. V. G. Marikhin, “On a two-dimensional Schrцdinger equation with a magnetic field with an additional quadratic integral of motion”, JETP Letters, 94:3 (2011), 243–247  mathnet  crossref  isi
    6. V. G. Marikhin, “On the two-dimensional classical motion of a charged particle in an electromagnetic field with an additional quadratic integral of motion”, JETP Letters, 97:7 (2013), 425–428  mathnet  crossref  crossref  isi  elib  elib
    7. V. G. Marikhin, “Quasi-Stäckel systems and two-dimensional Schrödinger equations in an electromagnetic field”, Theoret. and Math. Phys., 177:1 (2013), 1352–1360  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    8. A. V. Bolsinov, A. V. Borisov, I. S. Mamaev, “Geometrizatsiya teoremy Chaplygina o privodyaschem mnozhitele”, Nelineinaya dinam., 9:4 (2013), 627–640  mathnet
    9. Yehia H.M., Elmandouh A.A., “Integrable 2D Time-Irreversible Systems with a Cubic Second Integral”, Adv. Math. Phys., 2016, 8958747  crossref  mathscinet  zmath  isi  elib  scopus
    10. Agapov S.V., Bialy M., Mironov A.E., “Integrable Magnetic Geodesic Flows on 2-Torus: New Examples via Quasi-Linear System of PDEs”, Commun. Math. Phys., 351:3 (2017), 993–1007  crossref  mathscinet  zmath  isi  scopus
    11. Bolsinov A., Matveev V.S., Miranda E., Tabachnikov S., “Open Problems, Questions and Challenges in Finite-Dimensional Integrable Systems”, Philos. Trans. R. Soc. A-Math. Phys. Eng. Sci., 376:2131 (2018), 20170430  crossref  isi  scopus
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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