First, let me take as the definition of a Lagrangian the quantity that when put into the Euler Lagrange equations, it gives the correct equation of motion. It sounds like we need to know the equations of motion first. For example. the Lagrangian for a particle subject to a constant magnetic field. It is not your standard L=T-U. 1. With this in mind, when I write down T-U for a system, how do I know if it is also the Lagrangian of a system? 2. Also, this seems somewhat circular as to get the equations of motion we use the Lagrangian, but the Lagrangian is defined by the correct equations of motion. Can someone clarify this for me?