Seminar talk, 20 September 2023
Speaker: Konstantin Druzhkov
Title: Internal Lagrangians as variational principles
The principle of stationary action deals with Lagrangians defined on jets. However, for some reason, the intrinsic geometry of the corresponding equations knows about their variational nature. It turns out that the explanation is quite simple: each stationary-action principle reproduces itself in terms of the intrinsic geometry. More precisely, each admissible Lagrangian gives rise to a unique integral functional defined on some particular class of submanifolds of the corresponding equations. Such submanifolds can be treated as almost solutions since (informally speaking) they are composed of initial-boundary conditions lifted to infinitely prolonged equations. Intrinsic integral functionals produced by variational principles are related to so-called internal Lagrangians. This relation allows us to introduce the notion of stationary point of an internal Lagrangian, formulate the corresponding intrinsic version of Noether's theorem, and discuss the nondegeneracy of presymplectic structures of differential equations. Despite the generality of the approach, its application to gauge theories proves to be challenging. Perhaps the construction needs some modification in this case.