Two-Dimensional Hydrodynamics as a Class of Special Hamiltonian Systems

  • Kostyantyn M. Kulyk Institute for Single Cristals, Nat. Academy of Science Ukraine, Kharkiv, Ukraine
  • Vladimir V. Yanovsky Institute for Single Cristals, Nat. Academy of Science Ukraine, Kharkiv, Ukraine; V.N. Karazin Kharkiv National University, Kharkiv, Ukraine
Keywords: Hamiltonian, Lagrangian, Exact solutions, Two-dimensional hydrodynamics, Phase flow


The paper defines a class of Hamiltonian systems whose phase flows are exact solutions of the two-dimensional hydrodynamics of an incompressible fluid. The properties of this class are considered. An example of a Lagrangian one-dimensional system is given, which after the transition to the Hamiltonian formalism leads to an unsteady flow, that is, to an exact solution of two-dimensional hydrodynamics. The connection between these formalisms is discussed and the Lagrangians that give rise to Lagrangian hydrodynamics are introduced. The obtained results make it possible to obtain accurate solutions, such as phase flows of special Hamiltonian systems.


Download data is not yet available.


P.J. Morrison, ”Hamiltonian description of the ideal fluid,” Review of Modern Physics, 70(2), 467-521 (1998).

R. Salmon, ”Hamiltonian fluid mechanics,” Ann. Rev. Fluid Mech, 20, 225-256 (1988).

V.E. Zakharov, ”The Hamiltonian Formalism for waves in nonlinear media having dispersion,” Radiophys. Quantum Electron. 17, 326-343 (1974).

V.E. Zakharov, and E.A. Kuznetsov, ”Hamiltonian formalism for nonlinear waves,” Phys. Usp. 40, 1087 (1997).

C.S. Gardner, ”The Korteweg-de Vries equation and generalization I. The Korteweg-de Vries equation as a Hamiltonan system,” J. Math. Phys. 12(8), 1548-1551 (1971).

D. Serre, ”Invariants et degenerescence symplectique de l’equation d’Euler des fluids parfaits incompressibles,” C.R. Acad. Sci. Paris. Ser. A, 298, 349-352 (1984).

A.V. Tur, and V.V. Yanovsky, ”Invariants in Dissipationless hydrodynamics media”, J. Fluid. Mech. 248, 67-106 (1993).

D.G. Ebin, and J.E. Marsden, ”Groups of diffeomorphisms and the solution of the classical Euler equations for a perfect fluid,” Bull. Amer. Math. Soc. 75(5), 962-967 (1969).

D.G. Ebin, and J.E. Marsden, ”Groups of diffeomorphisms and the notion of an incompressible fluid,” Ann. of Math. Second Series, 92(1), 102-163 (1970).

B. Khesin, and R. Wendt, The Geometry of Infinite-Dimensional Groups, (Springer Berlin, Heidelberg, 2008).

S.P. Novikov, ”The Hamiltonian formalism and a many valued analog of Morse theory,” Russ. Math. Surveys, 37(5), 1-56 (1982).

A. Maltsev, and S. Novikov, ”Poisson Brackets of Hydrodynamic Type and Their Generalizations,” Journal of Experimental and Theoretical Physics, 132, 645-657 (2021).

O.I. Mokhov, ”Symplectic and Poisson structures on loop spaces of smooth manifolds, and integrable systems,” Russ. Math. Surv. 53, 515-623 (1998).

E.V. Ferapontov, ”Differential geometry of nonlocal Hamiltonian operators of hydrodynamic type,” Funct. Anal. Its Appl. 25, 195-204 (1991).

V.V. Kozlov, ”Hydrodynamics of Hamiltonian systems,” Vestnik Moskovskiogo Universiteta, Seriia 1: Matematika, Mekhanika, Nov.-Dec. 10-22 (1983). (in Russian)

V.E. Zakharov, ”The algebra of integrals of motion of two-dimensional hydrodynamics in clebsch variables,” Funct. Anal. Its Appl. 23, 189-196 (1989).

How to Cite
Kulyk, K. M., & Yanovsky, V. V. (2024). Two-Dimensional Hydrodynamics as a Class of Special Hamiltonian Systems. East European Journal of Physics, (2), 134-141.