Classical Mechanics

I have just realized that I never complete what I set out to do a couple months ago; to tell you about each one of my classes were about. Since I'm sure you are so excited about the prospect of reading about physics, I will enlighten you on yet another one of the courses, classical mechanics.

Classical mechanics is basically what people learn in the introductory physics classes, except in a new formulation. First off, some people may be wondering, well, what makes the physics "classical". The short answer is that classical physics is done on scales that render quantum effect negligible. For instance, the dynamics of planetary bodies or spinning tops are such that any quantum effects are suppressed and we can analyse the systems ignoring quantum contributions.

For this class we used two textbooks, of which, one is a classic, having been used for practically every graduate level mechanics course since it was written. The two textbooks were as follows:

1) Theoretical Mechanics of Particles and Continua - Fetter and Walecka
2) Classical Mechanics (3rd ed) - Herbert Goldstein et al. (the classic)

I must say, that we actually did not utilize Goldstein much, except as a supplement to our readings and to provide a clear understanding of material that Fetter and Walecka (FW) were brief on. FW had a good organization to it, the material was presented in good order, but they were too terse most times and their problems were ridiculously hard given their description of the material. Goldstein on the other hand, was almost too detailed, going off on numerous tangents, but it's a very complete text with problems that range from moderately easy to "wtf hard".

The material presented in the course began with Central Force problems (planetary orbits, particle scattering) presented in a fashion that was note too much different than introductory physics in FW. FW's chapter 1 was Goldstein's chapter 3, but Goldstein introduced the Lagrangian formulation in chapters 1 and 2 while that was left for FW's chapter 3. So here is how the material progressed with chapter numbers indicated:

A) Central Force (FW 1, G 3)
B) Accelerating Reference Frames (FW 2, G 4)
C) Lagrangian Formulation (FW 3, G 1+2)
D) Small Oscillations (FW 4, G 6)
E) Rigid Body Dynamics (FW 5, G 5)
F) Canonical Transformations and Hamiltonian Dynamics (FW 6, G 8+9+10)
G) Canonical Perturbation Theory (G 12)

Basically, most of the course is focused, as is the undergraduate classical mechanics course, on the lagrangian formulation of classical mechanics. Whereas in intro. physics we utilize force diagrams and compare forces using the Newtonian formulation (F = m*a), we utilize the energy of the system and define what is called a Lagrangian (L = T - V) were T is the kinetic energy of the system and V is the potential. Now this can be done in any coordinate system, but we always wish to transform to generalized coordinates, so instead of an x-y-z system, it may be easier to utilize spherical polar coordinates to describe the system (r-theta-phi). It can be shown that (and I will show you if you'd like) using an equation known as the Euler-Lagrange equation that the Lagrangian leads to Newton's second law immediately with little fuss over forces. The biggest downfall is that friction becomes a bear to work with so most all examples and problems are frictionless, but the mathematics is much easier to work with than is Newton's formulation for complex systems.

Is it really that much better? Well, here is an example that would be a bear in Newtonian mechanics. This is also an easy example off of my mechanics final.

A bead of mass M is threaded on a hoop, also of mass M and is free to move about the hoop. The hoop is attached to a bearing system that allows the hoop to swing like a pendulum. If we tap the system it goes in motion. Describe the motion of the bead on the hoop.

With Newtonian mechanics this would suck. With Lagrangian formulation it's really straight forward.

We also focused on the Hamiltonian formulation, which is just another way to formulate mechanics. It actually does not really make solving the problem any easier, it's just easier to work with at times since you try and make all of your variables constant using canonical transforms. No new physics involved, just a mathematical construct.

The course was really interesting, I particularly enjoyed the Hamiltonian formulation of mechanics and the subsequent Hamilton-Jacobi theory along with action-angle variables. Might not mean much, but it was fun and interesting to me.