First-order Hamiltonian field theory and mechanics

This article deals with the geometric analysis of the evolutionary and the polysymplectic approach in first-order Hamiltonian field theory. Based on a variational formulation in the Lagrangian picture, two possible counterparts in a Hamiltonian formulation are discussed. The main difference between these two approaches, which are important for the application, is besides a different bundle construction, the different Legendre transform as well as the analysis of the conserved quantities. Furthermore, the role of the boundary conditions in the Lagrangian and in the Hamiltonian pictures will be addressed. These theoretical investigations will be completed by the analysis of several examples, including the wave equation, a beam equation and a special subclass of continuum mechanics in the presented framework.