Classical Mechanics

by landonlehman

Today I finished reading section 25 (\S 25) in Mechanics by Landau and Lifshitz. There are 50 sections total, so I’m feeling good about my goal of finishing the book by the end of this summer.  I started off doing the problems (or at least making sure I understood each one), but that was taking too much time and I really wanted to focus on making progress on the reading, so I stopped. When the reading is done, I hope to return to the problems.

One of the challenging aspects of the problems is the assumption that doing any sort of elementary integral will be second nature.  Now, I could do most of the integrals in the problems, but some of them took about a page of trying to figure out the right trig substitution.  The work was good practice, but I didn’t feel like I was really learning more about mechanics.

Classical mechanics isn’t used all that much in day-to-day research in physics, so one might wonder why it is good to spend time learning it.  There are a couple of good reasons that I can think of off the top of my head.  First, it is a required course in most graduate programs in physics.  Therefore the people who currently have PhDs in physics and in a sense can control the curriculum still think it is required knowledge and perhaps they might know what they’re talking about.  Second, there are many applications of classical mechanics which prove useful in diverse areas of physics.

To back up the second reason, I can give an example.  Back when I was still working through the problems in Mechanics, I worked on Problem 3 at the end of \S 15.  This problem deals with adding a perturbing potential to the normal expression for the potential in the Kepler problem.  In the presence of such a perturbing potential, the perihelion of the orbit undergoes precession.  The problem in Mechanics calculates the amount of precession through each period. There is a similar problem in Goldstein (Problem 21, Ch. 3, 3rd ed.).  As I worked through the problem in Mechanics, I thought that it would be interesting to compare the amount of precession that would be caused by different functional forms of the perturbing potential.  Specifically, it would not be too hard to write a Mathematica notebook to do the integrals for power-law potentials (I did this).  I was interesting in learning more about this, with the idea in the back of my mind of maybe writing a paper on it (after a few years of study, of course).  So I looked it up, and discovered that the paper has already been written (“Orbital precession due to central-force perturbations,” by Adkins and McDonnell).  It was published in Physical Review D in 2007, and actually obtains general analytical results for power law and logarithmic perturbing potentials.  One of the useful things about this type of result is that it can be used to place constraints on the existence of new types of forces that might show up in theories such as extensions of general relativity.

So while learning classical mechanics it is possible to come up with ideas that could lead to the deeper exploration of some parts of physics and might even lead to publishable results.  And these results can be useful in furthering current research in areas of physics outside of classical mechanics.

While writing this, I thought of a couple more good reasons to study classical mechanics, besides the two just discussed.  But it is time to get some sleep, so perhaps I will discuss the next two reasons in another post.