On the compatible weakly nonlocal Poisson brackets of hydrodynamic type.
Maltsev, Andrei Ya. (2002)
International Journal of Mathematics and Mathematical Sciences
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Maltsev, Andrei Ya. (2002)
International Journal of Mathematics and Mathematical Sciences
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Praught, Jeffery, Smirnov, Roman G. (2005)
SIGMA. Symmetry, Integrability and Geometry: Methods and Applications [electronic only]
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Simonetta Abenda, Tamara Grava (2005)
Annales de l’institut Fourier
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We derive the modulation equations (Whitham equations) for the Camassa-Holm (CH) equation. We show that the modulation equations are hyperbolic and admit a bi-Hamiltonian structure. Furthermore they are connected by a reciprocal transformation to the modulation equations of the first negative flow of the Korteweg de Vries (KdV) equation. The reciprocal transformation is generated by the Casimir of the second Poisson bracket of the KdV averaged flow. We show that...
Dong Eui Chang, Anthony M. Bloch, Naomi E. Leonard, Jerrold E. Marsden, Craig A. Woolsey (2010)
ESAIM: Control, Optimisation and Calculus of Variations
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The purpose of this paper is to show that the method of controlled Lagrangians and its Hamiltonian counterpart (based on the notion of passivity) are equivalent under rather general hypotheses. We study the particular case of simple mechanical control systems (where the underlying Lagrangian is kinetic minus potential energy) subject to controls and external forces in some detail. The equivalence makes use of almost Poisson structures (Poisson brackets that may fail to satisfy the Jacobi...