Variational symmetries and Lagrangian multiforms

By considering the closure property of a Lagrangian multiform as a conservation law, we use Noether’s theorem to show that every variational symmetry of a Lagrangian leads to a Lagrangian multiform. In doing so, we provide a systematic method for constructing Lagrangian multiforms for which the closure property and the multiform Euler–Lagrange (EL) both hold. We present three examples, including the first known example of a continuous Lagrangian 3-form: a multiform for the Kadomtsev–Petviashvili equation. We also present a new proof of the multiform EL equations for a Lagrangian k-form for arbitrary k.