Poisson bracket on 1-forms and evolutionary partial differential equations

We introduce a bracket on 1-forms defined on J ∞ (S 1,R n), i.e. 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 a compatible flat connection (M,∇, o) are Hamiltonian in a generalized sense. Moreover, we show that if a metric g compatible with ∇ is also invariant with respect to o, then this generalized Hamiltonian setup reduces to the standard one. © 2012 IOP Publishing Ltd.