Find the Lagrangian from given equations of motion

Question

Is there a systematic procedure to generally obtain an appropriate Lagrangian that corresponds to any given equations of motion (if I know that it exists)?

Answer 1

In general, this is difficult, as the same dynamics can be written in many different forms.

In concrete cases, I'd do one of the following:

1. Work out the Hamiltonian (i.e., look for conserved quantities of a reasonably simple form), then work out pairs of conjugate variables that allow you to write the equation of motion in Hamiltonian form, then invert the canonical formalism to get the Lagrangian.

2. Write down the most general combinations of terms whose functional derivatives resemble those in the given equation, and then try to match terms, accounting for a possible (but assumed simple) integrating factor.