ABSTRACT
One may ask again, what is the point of using the Lagrangian approach rather than directly using classical equations? When we use the proper Lagrangian in classical mechanics we obtain, from Lagrange’s equation, Newton’s principles and the appropriate trajectory in a gravitational field. If we apply Lagrange’s equation to the appropriate Lagrangian for a charge in an electromagnetic field, we obtain Lorentz force. If we used the Lagrangian proposed in Chapter V for electromagnetic fields, we obtain Maxwell’s equations. We can introduce relativity by defining an appropriate Lagrangian. There are many other examples that we could introduce in other subfields of physics where the analysis could be made through the use of the Lagrangian approach. The exercise appears to be in the finding of the appropriate Lagrangian, which, when used with Lagrange’s equation, leads to the laws that we already know. In general, it is observed that the form of the Lagrangian that is required for the results to be compatible with already known laws is not obvious at all. The Lagrangian correctness is essentially demonstrated a posteriori.
