Poisson bracket on 1-forms and evolutionary partial differential equations

We introduce a bracket on 1-forms defined on J∞(S¹, Rⁿ), 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, ∇, ∘) are Hamiltonian in a generalized sense. Moreover, we show that if a metric g compatible with ∇ is also invariant with respect to ∘, then this generalized Hamiltonian set-up reduces to the standard one.