Maxwell-Klein-Gordon system with nontrivial coupling on four dimensional Minkowski spacetime

In this paper, we study about Maxwell-Klein-Gordon system with non-trivial coupling in four dimensional Minkowski spacetime with potential turned on. We start from Lagrangian of non-trivially coupled Maxwell-Klein-Gordon, then we derive the equations of motion and energy of the system. The coupling and potential function is chosen such that the Lagrangian is gauge invariant. Using Coulomb gauge and conservation of energy, we prove some inequality for energy which will be important to proving the existence of solution.