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Regul. Chaotic Dyn., 2015, Volume 20, Issue 1, Pages 74–93 (Mi rcd62)  

This article is cited in 19 scientific papers (total in 19 papers)

Simultaneous Separation for the Neumann and Chaplygin Systems

Andrey V. Tsiganov

St. Petersburg State University, ul. Ulyanovskaya 1, St. Petersburg, 198504, Russia

Abstract: The Neumann and Chaplygin systems on the sphere are simultaneously separable in variables obtained from the standard elliptic coordinates by the proper Bäcklund transformation. We also prove that after similar Bäcklund transformations other curvilinear coordinates on the sphere and on the plane become variables of separation for the system with quartic potential, for the Hénon-Heiles system and for the Kowalevski top. This allows us to speak about some analog of the hetero Bäcklund transformations relating different Hamilton–Jacobi equations.

Keywords: bi-Hamiltonian geometry, Bäcklund transformations, separation of variables

Funding Agency Grant Number
Russian Foundation for Basic Research 13-01-00061
Saint Petersburg State University 11.38.664.2013
This work was partially supported by RFBR grant 13-01-00061 and SPbU grant 11.38.664.2013.


DOI: https://doi.org/10.1134/S1560354715010062

References: PDF file   HTML file

Bibliographic databases:

MSC: 37K35, 53D22, 70H06
Received: 28.10.2014
Accepted:24.11.2014
Language:

Citation: Andrey V. Tsiganov, “Simultaneous Separation for the Neumann and Chaplygin Systems”, Regul. Chaotic Dyn., 20:1 (2015), 74–93

Citation in format AMSBIB
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\by Andrey V. Tsiganov
\paper Simultaneous Separation for the Neumann and Chaplygin Systems
\jour Regul. Chaotic Dyn.
\yr 2015
\vol 20
\issue 1
\pages 74--93
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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. A. V. Tsyganov, “Razdelenie peremennykh dlya odnogo obobscheniya sistemy Chaplygina na sfere”, Nelineinaya dinam., 11:1 (2015), 179–185  mathnet  elib
    2. Andrey V. Tsiganov, “Killing Tensors with Nonvanishing Haantjes Torsion and Integrable Systems”, Regul. Chaotic Dyn., 20:4 (2015), 463–475  mathnet  crossref  mathscinet  zmath  adsnasa  elib
    3. A. V. Tsiganov, “On auto and hetero Backlund transformations for the Henon-Heiles systems”, Phys. Lett. A, 379:45-46 (2015), 2903–2907  crossref  mathscinet  zmath  isi  scopus
    4. A. V. Tsiganov, “On the Chaplygin system on the sphere with velocity dependent potential”, J. Geom. Phys., 92 (2015), 94–99  crossref  mathscinet  zmath  isi  scopus
    5. A. V. Tsiganov, “Two integrable systems with integrals of motion of degree four”, Theoret. and Math. Phys., 186:3 (2016), 383–394  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    6. A. V. Tsiganov, “Abel's theorem and Bäcklund transformations for the Hamilton–Jacobi equations”, Proc. Steklov Inst. Math., 295 (2016), 243–273  mathnet  crossref  crossref  mathscinet  isi  elib
    7. Yu. A. Grigorev, A. P. Sozonov, A. V. Tsyganov, “Ob odnoi integriruemoi sisteme na ploskosti s potentsialom, zavisyaschim ot skorosti”, Nelineinaya dinam., 12:3 (2016), 355–367  mathnet  crossref  mathscinet  elib
    8. A. V. Tsiganov, “On a family of Backlund transformations”, Dokl. Math., 93:3 (2016), 292–294  crossref  crossref  mathscinet  zmath  isi  elib  scopus
    9. Andrey V. Tsiganov, “Bäcklund Transformations for the Nonholonomic Veselova System”, Regul. Chaotic Dyn., 22:2 (2017), 163–179  mathnet  crossref
    10. A. V. Tsyganov, “Ob odnoi integriruemoi sisteme na ploskosti s integralom dvizheniya shestoi stepeni po impulsam”, Nelineinaya dinam., 13:1 (2017), 117–127  mathnet  crossref  elib
    11. Andrey V. Tsiganov, “Integrable Discretization and Deformation of the Nonholonomic Chaplygin Ball”, Regul. Chaotic Dyn., 22:4 (2017), 353–367  mathnet  crossref
    12. A. V. Tsiganov, “Bäcklund transformations for the Jacobi system on an ellipsoid”, Theoret. and Math. Phys., 192:3 (2017), 1350–1364  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    13. A. V. Tsiganov, “New bi-Hamiltonian systems on the plane”, J. Math. Phys., 58:6 (2017), 062901  crossref  mathscinet  zmath  isi  scopus
    14. A. V. Tsiganov, A. P. Sozonov, “On auto and hetero Backlund transformations for the Henon-Heiles systems”, Proceedings of the Eighteenth International Conference on Geometry, Integrability and Quantization, eds. I. Mladenov, G. Meng, A. Yoshioka, Inst Biophysics & Biomedical Engineering Bulgarian Acad Sciences, 2017, 241–251  crossref  mathscinet  isi
    15. A. V. Tsiganov, “Backlund transformations and divisor doubling”, J. Geom. Phys., 126:SI (2018), 148–158  crossref  mathscinet  zmath  isi  scopus
    16. A. V. Tsiganov, “Duffing Oscillator and Elliptic Curve Cryptography”, Nelineinaya dinam., 14:2 (2018), 235–241  mathnet  crossref  elib
    17. A. V. Tsiganov, “Discretization of Hamiltonian systems and intersection theory”, Theoret. and Math. Phys., 197:3 (2018), 1806–1822  mathnet  crossref  crossref  adsnasa  isi  elib
    18. Andrey V. Tsiganov, “On Discretization of the Euler Top”, Regul. Chaotic Dyn., 23:6 (2018), 785–796  mathnet  crossref
    19. A. V. Tsiganov, “On exact discretization of cubic-quintic Duffing oscillator”, J. Math. Phys., 59:7 (2018), 072703  crossref  mathscinet  zmath  isi  scopus
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