Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 3 (2007), 062, 14 pages      math-ph/0612048      https://doi.org/10.3842/SIGMA.2007.062
Contribution to the Vadim Kuznetsov Memorial Issue

Weakly Nonlocal Hamiltonian Structures: Lie Derivative and Compatibility

Artur Sergyeyev
Mathematical Institute, Silesian University in Opava, Na Rybnícku 1, 746 01 Opava, Czech Republic

Received December 15, 2006, in final form April 23, 2007; Published online April 26, 2007

Abstract
We show that under certain technical assumptions any weakly nonlocal Hamiltonian structure compatible with a given nondegenerate weakly nonlocal symplectic structure J can be written as the Lie derivative of J −1 along a suitably chosen nonlocal vector field. Moreover, we present a new description for local Hamiltonian structures of arbitrary order compatible with a given nondegenerate local Hamiltonian structure of zero or first order, including Hamiltonian operators of the Dubrovin-Novikov type.

Key words: weakly nonlocal Hamiltonian structure; symplectic structure; Lie derivative.

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