Differential Geometry

I have been enjoying the last couple of days preparing for my Statistical Mechanics class. My new angle is to throw in a bit of differential geometry to expose students to its beauty as well as to give them a new and powerful tool to solve problems.

This morning, I was going over my notes on canonical transformations - a topic that I also covered in Classical Mechanics last semester. One concept that I still find a tad uncomfortable is in the manipulations of the quantities q, the coordinate, and q dot, its associated velocity. Central to the calculus of variations is the assumption that these two variables are independent. I can logically understand how this is the case, but my gut protests.

One of my colleagues, who shares an interest in the beauty of differential geometry, gave me the book Applied Differential Geometry, by William L. Burke. I noticed recently the most magnificent dedication on its inner cover, which reads, "To all those who, like me, have wondered how in the hell you can change q dot without changing q."

I can't wait to dive into this wonderful book; but, it will have to wait until I meet a looming proposal deadline and have prepared/delivered an invited talk in a few days. I must get back to preparing for my class, which meets in 45 minutes. Mornings before class are one of my favorite times...