Trivial Lagrangians in field theory
Let \(Y\) be an \((n+m)\)-dimensional fibered manifold over an \(n\)-dimensional base space \(X\) with projection \(\pi: Y\to X\). Let the \(r\)-jet prolongation of \(Y\) be denoted by \(J^rY\), \(\pi^r: J^rY\to X\) and \(\pi^{r,s}: J^rY\to J^sY\), \(r\geq s\geq 0\), being the corresponding canonical projections. The main result of the paper describes the local structure of trivial Lagrangians of order \(r\) and asserts the following. Theorem. A Lagrangian \(\lambda\) of order \(r\) over \((Y,\pi,X)\) is trivial if and only if for each point \(y\in Y\) there exists a fibered chart \((V, \psi)\) on \(Y\) and an \((n-1)\)-form \(\chi\) on \(V^{r-1}\subset J^{r-1}Y\) such that \(\lambda=hd\chi\) on \(V^r\). Here \(h\) denotes the horizontal component and for an open set \(V\subset Y\), \(V^r=(\pi^{r,0})^{-1}(V)\). This generalizes a 25 year old result of the first author [\textit{D. Krupka}, Arch. Math., Brno 10(1974), 55--61 (1975; Zbl 0337.33012)].
scientific article; zbMATH DE number 1308137
scientific article