Internal Lagrangians and gauge systems

In classical mechanics, the Hamiltonian formalism is given in terms of instantaneous phase spaces of mechanical systems. This explains why it can be interpreted as an encapsulation of the Lagrangian formalism into the intrinsic geometry of equations of motion. This observation can be generalized to the case of arbitrary variational equations. To do this, we describe instantaneous phase spaces using the intrinsic geometry of PDEs. The description is given by the lifts of involutive codim-1 distributions from the base of a differential equation viewed as a bundle with a flat connection (Cartan distribution). Such lifts can be considered differential equations, which one can regard as gauge systems. They encode instantaneous phase spaces. In addition, each Lagrangian of a variational system generates a unique element of a certain cohomology of the system. We call such elements internal Lagrangians. Internal Lagrangians can be varied within classes of paths in the instantaneous phase spaces. This fact yields a direct (non-covariant) reformulation of the Hamiltonian formalism in terms of the intrinsic geometry of PDEs. Finally, the non-covariant internal variational principle gives rise to its covariant child.