On the variational principle in the unfolded dynamics

TL;DR

This paper explores the relationship between off-shell and on-shell unfolded systems, introducing invariant constraints and a cohomological framework to understand their dynamics and conditions for being derived from a Lagrangian.

Contribution

It develops a new formulation for invariant on-shell constraints in unfolded dynamics, extending the Q-derivation to a bicomplex and establishing conditions for Euler-Lagrange equations.

Findings

Introduces a bicomplex extension of Q-derivation.

Defines invariant on-shell constraints in cohomological terms.

Proves conditions for constraints to be Euler-Lagrange equations.

Abstract

The interplay between off-shell and on-shell unfolded systems is analysed. The formulation of invariant constraints that put an off-shell system on shell is developed by adding new variables and derivation in the target space, that extends the original $Q$-derivation of the unfolded system to a bicomplex. The analogue of the Euler-Lagrange equations in the unfolded dynamics is suggested. The general class of invariant on-shell equation constraints is defined in cohomological terms. The necessary and sufficient condition for the on-shell equation constraints being Euler-Lagrange for some Lagrangian system is proven. The proposed construction is illustrated by the scalar field example.