Covariant Hamiltonian formalism for the calculus of variations with several variables: Lepage-Dedecker versus De Donder-Weyl (Q1772141)

The authors develop a covariant Hamiltonian formalism for the calculus of variations with several variables (field theories), by using differential geometric methods. More precisely, this paper is devoted to the study of Lepage-Dedecker theory, which is compared with the more usual De Donder-Weyl theory. Both formalisms are based on the same geometrical object: a multisymplectic manifold. But if many contributions to De Donder-Weyl theory construct multisymplectic manifolds having the same dimension as the ones in Lagrangian formalism, when dealing with Lepage-Dedecker theories, one is forced to work with multisymplectic manifolds whose dimension is larger than the number of physical variables. Therefore, Lepage theories are more complicated to deal with. The advantage of Lepage-Dedecker theories is that no additional geometrical structures, such as connections, are needed and, in particular, in the setting by the authors the Hamiltonian is a global function defined on the multisymplectic manifold. Another nice property of this theory is related to the Legendre correspondence replacing Legendre transform in De Donder-Weyl approach. This correspondence behaves differently, ignoring the singularities whenever the Lagrangian is degenerate.