Fibered manifold; Lagrangian; variational sequence; contact - form, contact symmetry;Helmholtz form

One of the results of the variational sequence theory, related to the inverse problem of the calculus of variations, states that a dynamical form $\varepsilon$, representing a system of partial differential equations, is locally variational if and only if the Helmholtz form $H(\varepsilon)$ vanishes. In this paper, a relationship between the Lie derivatives of $\varepsilon$ and $H(\varepsilon)$ is studied. It is shown that invariance of the Helmholtz form $H(\varepsilon)$ with respect to a vector field $Z$ preserving contact forms is equivalent with local variationality of the Lie derivative of $\varepsilon$ by $Z$.

One of the results of the variational sequence theory, related to the inverse problem of the calculus of variations, states that a dynamical form $\varepsilon$, representing a system of partial differential equations, is locally variational if and only if the Helmholtz form $H(\varepsilon)$ vanishes. In this paper, a relationship between the Lie derivatives of $\varepsilon$ and $H(\varepsilon)$ is studied. It is shown that invariance of the Helmholtz form $H(\varepsilon)$ with respect to a vector field $Z$ preserving contact forms is equivalent with local variationality of the Lie derivative of $\varepsilon$ by $Z$.