# Variational symmetries and Lagrangian multiforms

## TL;DR

This paper establishes a systematic link between variational symmetries and Lagrangian multiforms using Noether's theorem, introduces a method for constructing multiforms with closure properties, and provides new examples including a Lagrangian 3-form for KP.

## Contribution

It provides a systematic method to construct Lagrangian multiforms from variational symmetries and introduces the first known Lagrangian 3-form example for KP.

## Key findings

- Derived a systematic construction method for Lagrangian multiforms.
- Presented the first known Lagrangian 3-form for the KP equation.
- Provided a new proof of multiform Euler-Lagrange equations for arbitrary k.

## Abstract

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) equations both hold. We present three examples, including the first known example of a 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.