See (Nishikawa 2002) about variational definition. Nishikawa, Seiki (2002). Variational problems in geometry. Translations of Mathematical Monographs; Iwanami Series in Modern Mathematics. Vol. 205. Translated by Abe, Kinetsu. Providence, R. I.: American Mathematical Society. ISBN0-8218-1356-0. ISSN0065-9282, translated from:{{cite book}}: CS1 maint: postscript (link)