Lagrangian and Hamiltonian formulation for infinite-dimensional systems - a variational point of view

In this article we use the Lagrange multiplier method, which is well-known in constrained optimization theory, to derive several different Hamiltonian counterparts to Lagrangian systems described by partial differential equations in a variational setting. The main observation is the fact that unconstrained, infinite-dimensional systems can be formulated as constrained variational problems, where the constraints are used to hide some or all derivative variables appearing in the Lagrangian. Depending on the chosen derivative variables that are affected by this approach, different representations of the same dynamical system can be achieved. These theoretical investigations will be applied to a demonstrative example from mechanics.