Equivalent Lagrangians and the Inverse Variational Problem with Applications

Abstract

We show how the study of the invariance of the functional in the variational problem is used for the construction of Lagrangians for some target equations from equations whose Lagrangians are known. Where the Lagrangian is not unique, we have the freedom to choose the Lagrangian that is most suitable (for e. g., one that admits the maximal Noether algebra of point symmetry generators). The method may be used in the multi dimensional case in the dependent and independent variables. A variety of examples with discussions are presented.