One of the key parts of any course in theoretical physics is the development of a wide range of procedures, which become more and more advanced, for treating problems in classical mechanics and dynamics. In one way or another, these are all extensions of the basic principles enunciated by Newton in the Principia, although some of them appear to bear only a distant resemblance to Newton's three laws of motion. As an example of the variety of ways in which the fundamentals of mechanics and dynamics can be expounded, here is a list of some of the different approaches which I found in the text book Foundations of Physics by R.B. Lindsay and H. Margenau, which I read with some trepidation as a first-year undergraduate:

• Newton's laws of motion;

• D'Alembert's principle;

• the principle of virtual displacements;

• Gauss's principle of least constraint;

• Hertz's mechanics;

• Hamilton's principle and the principle of least action;

• Generalised coordinates and the methods of Lagrange;

• the canonical equations of Hamilton;

• the transformation theory of mechanics and the Hamilton–Jacobi equations.

This is not the place to go into these different approaches – this is more than adequately covered in standard texts such as Goldstein's Classical Mechanics. Rather, in Chapter 7, I want to emphasise some of the features of these approaches which provide insights into different aspects of dynamical systems.

First of all, it is important to appreciate that these different approaches are fully equivalent.