On Hamiltonian Hierarchies Associated with Hyperbolic Euler Equations

Abstract

The canonical coordinate-momenta formalism with internal degrees of freedom for hyperbolic Euler–Lagrange equations is applied to the study of flows of Hamiltonian symmetries. The differential constraint between coordinates and momenta supplies the Hamiltonian structure for the initial hyperbolic system and endows the evolution of coordinates and induced evolution of momenta with the Hamiltonian operators which are mutually inverse. Two illustrative examples are given: the bi-Hamiltonian Korteweg–de Vries hierarchy of Noether symmetries of the wave equation is connected with the multicomponent modified KdV analogs associated with the two-dimensional Toda equations uxy e exp(Ku).