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Hamiltonian differential operators and contact geometry. (English. Russian original) Zbl 0635.58006

Funct. Anal. Appl. 21, No. 1-3, 217-223 (1987); translation from Funkts. Anal. Prilozh. 21, No. 3, 53-60 (1987).
The problem of classification of Hamiltonian differential operators is investigated. An operator (1) \(L=\sum^{N}_{i=0}a_ i(x,u,u_ x,...)(\frac{d}{dx})^ i\) is called the Hamiltonian if the formula \[ \{I,J\}=\int (\delta I/\delta u)L(\delta J/\delta u)dx \] defines a Poisson bracket on the space of all functionals \(H=\int h(x,u,u_ x,...)dx\). The purpose of the paper is to get some Darboux-type results to classify Hamiltonian operators of the first and the third order under contact Bäcklund transformations. The notion of a Hamiltonian pair of operators, i.e. \(L_ 1,L_ 2\) of the form (1), such that \(\lambda L_ 1+\mu L_ 2\) is a Hamiltonian operator for any \(\lambda\),\(\mu\in {\mathbb{R}}\), is very important in the theory of integrability of evolution equations. The author studies the problem of classification of such pairs, consisting of \(L_ 1\) having first order and \(L_ 2\) having third order.
Reviewer: I.Ya.Dorfman

MSC:

37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
58D05 Groups of diffeomorphisms and homeomorphisms as manifolds
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