Classical Mechanics
Motivation #
So you’ve taken a first year college course on mechanics. You know Newton’s three laws by heart, and you can draw free body diagrams in your sleep. You might think that you know all there is to know about the behavior of large, dry, slow-moving objects, and in a certain sense, you do. Given the force laws and the initial conditions, you can in principle always crank out the state of the system numerically. Sometimes when you’re lucky, you’ll even be able to find an exact solution.
The problem is that force methods are sometimes intractable when a pen and paper solution is possible. For instance, suppose I give you the initial position and velocity of a chain constrained by one or more frictionless surfaces moving only under the influence of gravity and ask you to solve for its future motion. Doing this with forces will be a disaster. For every infinitesimal chain link, you’ll have to draw a free body diagram and set up a vector equation with special cases for parts of the chain that are and are not in contact with surfaces. Then you apply continuity conditions (encoding the fact that the chain can’t break) and set up a huge second order differential equation. You probably won’t be able to solve it, and even if you do, your solution won’t be robust to changes in the constraints. (1)Eg, if you solve for the motion of the Mould fountain, you won’t necessarilly be able to solve for the motion of the Yokoyama fountain.
The good news is that there’s a better way. We can encode the force law and kinematic constraints in a function called the Lagrangian and solve for the motion of the system by minimizing that function’s time integral. Less vector juggling is required of us, and our solution will generalize more easily to problems with different constraints.
This is one of the goals of classical mechanics: to make hard physics problems easier by encoding them as optimization problems on functions.
Questions #
- How do we know that there couldn’t be a force that depended on the acceleration of a particle, or on higher derivatives of its position? Relativity…?
- What does Noether’s Theorem really say, and what are all of its assumptions? The loose statement I know—for every symmetry there has to be a conserved quantity—seems shockingly strong.
- Does deep learning raise the status of Lagrangian mechanics, as Alex Alemi suggests in this lecture?
Reading List #
Theory
- Landau & Lifshitz, Course of Theoretical Physics, Vol 1: Mechanics. I don’t tend to like Russian textbooks very much, but this one is beautiful.
- Woodhouse, Introduction to Analytical Mechanics
- Kibble and Berkshire, Classical Mechanics
- Morin, Introduction to Classical Mechanics chapter 6
- Feynman Lectures, lecture II.19 on the Principle of Least Action
Applications
- Denny, “A uniform explanation of all falling chain phenomena” in AJP.
Last updated 13 October 2024