Theoretical and Mathematical Physics

, Volume 193, Issue 3, pp 1725–1736 | Cite as

Hamiltonian operators in differential algebras

  • V. V. ZharinovEmail author


We develop a previously proposed algebraic technique for a Hamiltonian approach to evolution systems of partial differential equations including constrained systems and propose a defining system of equations (suitable for computer calculations) characterizing the Hamiltonian operators of a given form. We demonstrate the technique with a simple example.


differential algebra Lie–Poisson structure Jacobi identity Hamiltonian operator Hamiltonian evolution system 


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  1. 1.
    V. V. Zharinov, “Lie–Poisson structures over differential algebras,” Theor. Math. Phys., 192, 1337–1349 (2017).MathSciNetCrossRefGoogle Scholar
  2. 2.
    P. J. Olver, Applications of Lie Groups to Differential Equations, Springer, New York, (1993).CrossRefzbMATHGoogle Scholar
  3. 3.
    V. V. Zharinov, Lecture Notes on Geometrical Aspects of Partial Differential Equations, World Scientific, Singapore, (1992).CrossRefzbMATHGoogle Scholar
  4. 4.
    N. H. Ibragimov, Transformation Groups in Mathematical Physics [in Russian], Nauka, Moscow (1983); English transl.: Transformation Groups Applied to Mathematical Physics (Math. Its Appl. Sov. Ser., Vol, 3), Springer, Dordrecht (1985).zbMATHGoogle Scholar
  5. 5.
    V. V. Zharinov, “Evolution systems with constraints in the form of zero-divergence conditions,” Theor. Math. Phys., 163, 401–413 (2010).CrossRefzbMATHGoogle Scholar
  6. 6.
    V. V. Zharinov, “Bäcklund transformations,” Theor. Math. Phys., 189, 1681–1692 (2016).CrossRefzbMATHGoogle Scholar
  7. 7.
    I. M. Gel’fand and I. Ya. Dorfman, “Hamiltonian operators and algebraic structures related to them,” Funct. Anal. Appl., 13, 248–262 (1979).CrossRefzbMATHGoogle Scholar
  8. 8.
    B. A. Dubrovin and S. P. Novikov, “Hydrodynamics of weakly deformed soliton lattices: Differential geometry and Hamiltonian theory,” Russ. Math. Surveys, 44, 35–124 (1989).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    M. O. Katanaev, “Killing vector fields and a homogeneous isotropic universe,” Phys. Usp., 59, 689–700 (2016).ADSCrossRefGoogle Scholar
  10. 10.
    G. A. Alekseev, “Collision of strong gravitational and electromagnetic waves in the expanding universe,” Phys. Rev. D, 93, 061501 (2016).ADSMathSciNetCrossRefGoogle Scholar
  11. 11.
    A. P. Chugainova and V. A. Shargatov, “Stability of discontinuity structures described by a generalized KdV–Burgers equation,” Comput. Math. Math. Phys., 56, 263–277 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    A. G. Kulikovskii and A. P. Chugainova, “Study of discontinuities in solutions of the Prandtl–Reuss elastoplasticity equations,” Comput. Math. Math. Phys., 56, 637–649 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    V. A. Vassiliev, “Local Petrovskii lacunas close to parabolic singular points of the wavefronts of strictly hyperbolic partial differential equations,” Sb. Math., 207, 1363–1383 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    V. V. Kozlov, “On the equations of the hydrodynamic type,” J. Appl. Math. Mech., 80, 209–214 (2016).MathSciNetCrossRefGoogle Scholar
  15. 15.
    V. M. Buchstaber, “Polynomial dynamical systems and the Korteweg–de Vries equation,” Proc. Steklov Inst. Math., 294, 176–200 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    V. P. Pavlov and V. M. Sergeev, “Fluid dynamics and thermodynamics as a unified field theory,” Proc. Steklov Inst. Math., 294, 222–232 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    A. G. Kulikovskii and A. P. Chugainova, “A self-similar wave problem in a Prandtl–Reuss elastoplastic medium,” Proc. Steklov Inst. Math., 295, 179–189 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    A. T. Il’ichev and A. P. Chugainova, “Spectral stability theory of heteroclinic solutions to the Korteweg–de Vries–Burgers equation with an arbitrary potential,” Proc. Steklov Inst. Math., 295, 148–157 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    G. A. Alekseev, “Integrable and non-integrable structures in Einstein–Maxwell equations with Abelian isometry group G2,” Proc. Steklov Inst. Math., 295, 1–26 (2016).MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2017

Authors and Affiliations

  1. 1.Steklov Mathematical Institute of Russian Academy of SciencesMoscowRussia

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