ABSTRACT
We have gained considerable experience in setting up Newton’s equations of motion in a variety of problems. If the system is not subject to external constraints, the equations of motion are usually easy to set up in Cartesian coordinates. If either the system is subject to external constraints or Cartesian coordinates are not used, then the equations of motion may be difficult to solve or even to formulate. Lagrange found a way to circumvent this problem by the use of generalized coordinates qi (to be defined soon). In terms of these generalized coordinates, we could write the equations of motion in a form that is equally suitable for all coordinates. Furthermore, the introduction of generalized coordinates can take advantage of constraints on a dynamic system. It is generally a much more pressing matter to take care of the constraints imposed on the motion of the dynamic system. The existence of constraints gives rise to two difficulties. First, the coordinates of the dynamic system are connected by the equations of constraints, so they are not all independent. As a result, the equations of motion are also not all independent. Second, the forces of the constraints are usually very complex or unknown. Thus, we may find ourselves unable to write the equations of motion. In order to circumvent these difficulties, alternative formulations to Newtonian theory have been developed. Each is based on the idea of energy and is so constructed that Newtonian theory may be recovered from it. Moreover, each is expressed in terms of generalized coordinates. This chapter presents a discussion on Lagrangian dynamics, which were developed by Joseph-Louis Lagrange (1736-1813) and others.
