Variational Symmetries and Lie Reduction for Frobenius Systems of Even Rank

Abstract

Let $\mathcal{I}$ be Frobenius system of even rank. Consider a closed two-form $\Pi \in \mathcal{I} \bigwedge \mathcal{I}$ of maximal rank. A vector field $X$such that $L_X \Pi = 0$ is called a symmetry of $\Pi$. We determine the relationship between the solvable Lie group of symmetries of $\Pi$ and the rank of the reduced system obtained from $\mathcal{I}$ by Lie reduction. For an Euler–Lagrange system of ODE’s with the corresponding Lagrangian $L$, $\Pi$ can be taken to be the differential of the Poincaré–Cartan form $\eta_L$. A symmetry of $\Pi = \mathbf{d}\eta_L$ is a variational symmetry of the Lagrangian $L$. A proof of Noether’s theorem for Frobenius systems of even rank is provided.