Advanced Classical Mechanics/Introduction
Previous physics courses have presented many applications of Newton's second law where the force, , is given or is a combination of constraining forces and given ones. The downside was that each phenomenon has its own set of forces to remember and different constraints required different formulations.
Physics is about unifying principles.
- Can we unify these diverse forces into a single principle?
- The answer is yes, yes. We can find two equivalent principles (ways of looking at classical mechanics).
- But we get more than that. These principles with only slight modification apply to quantum mechanics, electrodynamics and general relativity - in other words to all of physics.
The first is The Principle of Least Action that we will cover near the beginning of the course. This principle is often the easiest route to finding and solving the equations of motion of a system with constraints. We can formulate electrodynamics and general relativity with the principle, and it also provides a powerful (but not so easy) route to quantum mechanics.
The second is Hamilton's Equations. This formulation provides the easiest route to quantum mechanics and also gives us the powerful tool of Phase Space.
To get to the principle of least action we shall start with (as in Schaum, Goldstein and Kibble). We could have given the principle as an axiom (as in Landau and Lifshiftz) and found as a consequence. Since you are already familiar with the first path is probably more comfortable.