Noether Symmetries of the Area-Minimizing Lagrangian

Abstract

It is shown that the Lie algebra of Noether symmetries for the Lagrangian minimizing an $(n-1)$-area enclosing a constant $n$-volume in a Euclidean space is $so(n){\oplus }_s{ℝ}^n$ and in a space of constant curvature the Lie algebra is $so(n)$. Furthermore, if the space has one section of constant curvature of dimension $n_1$, another of $n_2$, and so on to $n_k$ and one of zero curvature of dimension $m$, with $n\ge {\sum }_{j=1}^k{n}_j+m$ (as some of the sections may have no symmetry), then the Lie algebra of Noether symmetries is ${\oplus }_{j=1}^{k}so(n_j+1)\oplus (so(m){\oplus }_s{ℝ}^m)$.