Lagrangian and Hamiltonian duality

We propose a setting for De Donder–Hamilton field theory in jet bundles, generalizing the usual multisymplectic formalism. Using a reformulation of Hamilton theory for the family of local Lagrangians related to a global Euler–Lagrange form, we construct a dual Hamiltonian bundle and corresponding Legendre maps, linking a Lagrangian system on a jet bundle with a canonical Hamiltonian system on the affine dual. Our approach significantly extends the family of regular variational problems that can be treated directly within a dual Hamiltonian formalism, thus avoiding the necessity to use the Dirac constraint formalism.