On the derivation of fluxes for conservation laws in Hamiltonian systems

The use of Noether's theorem and Lie-group techniques provides a systematic method for investigating the conservation laws of evolution equations, that is, expressions of the form DtT + div F = 0. For Hamiltonian systems, the method gives only the density T, and the flux F must be computed indirectly using the evolution equations. Here, a direct procedure for calculating the flux is developed based on the density of the conserved quantity and on the associated symmetry generator. Simple expressions are given for specific classes of conservation laws. Particular attention is paid to Hamiltonian systems involving non-local operators.