Poisson bracket on 1-forms and evolutionary partial differential equations

We introduce a bracket on 1-forms defined on ${\cal J}^{\infty}(S^1, $\mathbb{R}$^n)$, the infinite jet extension of the space of loops and prove that it satisfies the standard properties of a Poisson bracket. Using this bracket, we show that certain hierarchies appearing in the framework of $F$-manifolds with compatible flat connection $(M,\mathrm{\nabla },\circ )$ are Hamiltonian in a generalized sense. Moreover, we show that if a metric $g$ compatible with $\mathrm{\nabla }$ is also invariant with respect to $\circ$, then this generalized Hamiltonian set-up reduces to the standard one.