Part 4/6:
This idea was met with fierce criticism and ridicule from Maupertuis' contemporaries. However, the mathematician Leonhard Euler saw the potential in the principle and worked to put it on firmer mathematical footing. He showed that the principle of least action only holds if the total energy is conserved.
It was left to Joseph-Louis Lagrange to provide a general proof of the principle of least action. Lagrange's key insight was to rewrite the principle in terms of the Lagrangian - the difference between kinetic and potential energy. This allowed him to derive the equations of motion for any system using a simple variational principle.