In this chapter, we introduce the “Lagrangian”, a function of the generalized coordinates and velocities that is a very powerful method for describing physical systems. We will see that this scalar function, \(L(q_1,\dots ,q_n,\dot{q}_1,\dots ,\dot{q}_n,t)\), contains all of the information that is required to describe a system. We will see that all of classical mechanics can be described using the Lagrangian and the methods of the calculus of variations. We will find that properties of the Lagrangian result in a much deeper understanding of conservation laws.