Lagrangians for differential equations of any order

In this work the inverse problem of the variational calculus for systems of differential equations of any order is analyzed. It is shown that, if a Lagrangian exists for a given regular system of differential equations, then it can be written as a linear combination of the equations of motion. The conditions that these coefficients must satisfy for the existence of an S-equivalent Lagrangian are also exhibited. A generalization is also made of the concept of Lagrangian symmetries and they are related with constants of motion.