On the Legendre transformation for a class of nonregular higher-order Lagrangian field theories

In the framework of higher-order calculus of variations, the generalized Legendretransformation for a wide class of Lagrangians is considered, which depend in a nonregular way on the derivatives of maximal order. A rigorous theory is discussed for Lagrangians depending on a constant rank set of affine combinations of these derivatives. This allows the reduction of the Poincaré–Cartan formalism and the Hamiltonian formalism to the appropriate constraint in the appropriate phase space of the problem. The case considered here covers many important physical examples, such as the Yang–Mills theories (at order one) and relativistic metric theories of gravitation (at order two).