  Generalised Hydrodynamics (GHD) describes the large-scale inhomogeneous
dynamics of integrable (or close to integrable) systems in one dimension of
space, based on a central equation for the fluid density or quasi-particle
density: the GHD equation. We consider a new, general form of the GHD equation:
we allow for spatially extended interaction kernels, generalising previous
constructions. We show that the GHD equation, in our general form and hence
also in its conventional form, is Hamiltonian. This holds also including force
terms representing inhomogeneous external potentials coupled to conserved
densities. To this end, we introduce a new Poisson bracket on functionals of
the fluid density, which is seen as our dynamical field variable. The total
energy is the Hamiltonian whose flow under this Poisson bracket generates the
GHD equation. The fluid density depends on two (real and spectral) variables so
the GHD equation can be seen as a $2+1$-dimensional classical field theory. We
further show the system admits an infinite set of conserved quantities that are
in involution for our Poisson bracket, hinting at integrability of this field
theory.
