Abstract
An expression is derived for the variation of Lagrangians which are such that the set of admissible variables of variation is star-shaped. If such a Lagrangian leads to identically vanishing Euler-Lagrange expressions then it is shown that under suitable circumstances the Lagrangian in question must be an ordinary divergence. Furthermore, an expression is given for the ‘vector’ field which appears in this ordinary divergence.
Similar content being viewed by others
Bibliography
Buchdahl, H. A.,On the Nonexistence of a Class of Static Einstein Spaces Asymptotic at Infinity to a Space of Constant Curvature, J. Math. Phys.1, 537–541 (1960).
Buchdahl, H. A.,On Functionally Constant Invariants of the Riemann Tensor, Proc. Cambridge Phil. Soc.68, 179–185 (1970).
Funk, P.,Variationsrechnung und ihre Anwendung in Physik und Technik (Springer, Berlin-Göttingen-Heidelberg, 1962).
Horndeski, G. W.,Dimensionally Dependent Divergences, Proc. Cambridge Phil. Soc.72, 77–82 (1972).
Horndeski, G. W.,Dimensionally Dependent Divergences and Local Parallelizations, Tensor21, 83–90 (1973).
Königsberger, L.,Die Principien der Mechanik für mehrere unabhängige Variable, J. Reine Angew. Math.124, 202–277 (1902).
Lovelock, D.,The Euler-Lagrange Expression and Degenerate Lagrange Densities, J. Austral. Math. Soc.14 (4), 482–495 (1972).
Rund, H.,Integral Formulae Associated with the Euler-Lagrange Operators of Multiple Integral Problems in the Calculus of Variations Aequationes Math.11, 212–229 (1974).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Horndeski, G.W. Sufficiency conditions under which a lagrangian is an ordinary divergence. Aeq. Math. 12, 232–241 (1975). https://doi.org/10.1007/BF01836551
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01836551