Scope of Course

The scope of this course is indicated by its title, ``Analytical Classical Dynamics''. Taking the elements of the title in reverse order, ``Dynamics'' is the study of the motions of the various objects in the world around us. A mathematical theory of dynamics is an axiomatic system, ultimately based on a few fundamental laws, which can be used to predict these motions. By ``Classical", we understand that the theory of motion which we are going to employ in our investigation of dynamics is that first published by Isaac Newton in 1687. We now know that this theory is only approximately true. The theory breaks down when the velocities of the objects under investigation approach the speed of light, and must be replaced by Einstein's special theory of relativity. The theory also breaks down on the atomic and subatomic scales, and must be replaced by quantum mechanics. In this course, we shall neglect both relativistic and quantum effects entirely. It follows that we must restrict our investigation to the motions of large (compared to an atom) slow (compared to the speed of light) objects. Fortunately, most of the motions which we observe in the world around us fall into this category. Finally, by ``Analytical'', we understand that we are primarily interested in those types of motion whose governing differential equations can be solved via standard analytic techniques. In practice, this means that the governing equations must be linear in nature, since our ability to solve nonlinear differential equations analytically is very limited. Fortunately, a wide range of the observed motions in the world around us are governed, either exactly or approximately, by linear differential equations. Unfortunately, there is one very interesting type of motion which is definitely not governed by linear differential equations--namely, chaotic motion. It is, in fact, impossible to make a meaningful investigation of chaotic motion without resorting to numerical methods for solving the associated differential equations. Hence, we shall postpone any discussion of chaotic motion until the end of this course.