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By considering the closure property of a Lagrangian multiform as aconservation law, we use Noether's theorem to show that every variationalsymmetry of a Lagrangian leads to a Lagrangian multiform. In doing so, weprovide a systematic method for constructing Lagrangian multiforms for whichthe closure property and the multiform Euler-Lagrange (EL) equations both hold.We present three examples, including the first known example of a Lagrangian3-form: a multiform for the Kadomtsev-Petviashvili equation. We also present anew proof of the multiform EL equations for a Lagrangian k-form for arbitraryk.

Variational symmetries and Lagrangian multiforms