Classical Mechanics

The big idea #

Encode physics problems as optimization problems on functions.

Newtonian mechanics is about turning physics problems into initial value problems. Given the force laws and the initial configuration of a system, you just crank out the evolution of the system. This is a fully general procedure. Any classical mechanics problem that you can solve with Lagrangians and Hamiltonians you can also solve with forces.

Why is it sometimes better to optimize a function rather than solving an IVP?

Vector calculus is fiddly and easy to mess up. The Newtonian method requires you to solve a differential vector equation, whereas the Lagrangian method encodes all relevant information in a single scalar function. The first approach gives you many more opportunities to make a small mistake.

I want to draw a loose analogy to special relativity. The Lorentz transformations are equivalent to the three fundamental effects: time dilation, length contraction, and the rear clock ahead effect. Either one can be derived from the other, but we sometimes prefer the Lorentz method because it gives you fewer chances to slip up. Once you’ve written down the ST coordinates of the events you care about, you just put them through the transformations, and you’re done. Plug and chug. No further judgement is required on your part.

Similarly, solving a classical mechanics problem becomes a rote exercise once you’ve written down $ L$ or $ H$ as the case may be. Take derivatives, solve, and put in the initial conditions. Plug and chug. You gain less physical insight than you would from solving N2, but you also run less risk of error.

Newtonian methods don’t generalize to quantum systems. In QM, the position of a particle is described by a wavefunction extending over space, so there’s no well-defined position for you to plug into a force law. It doesn’t make sense to apply N2 to such a particle, but it does make sense to define an operator $H$ that will satisfy (a generalized version of) the Hamilton-Jacobi Equation. The Lagrangian and Newtonian methods are equivalent within classical mechanics, but when we come to quantum mechanics, it turns out that one method has legs while the other doesn’t.

Newtonian methods don’t play nicely with relativity. The Lorentz transformation for a force is quite awkward and understanding how to apply it can be rather subtle. If we have to do relativistic dynamics, we’d much rather do them in Lagrangian terms, since the action is Lorentz invariant.

[TODO: Why is this true? What assumptions are needed to make the action Lorentz invariant?]

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?
• Can we say anything about how the computational complexity of solving the Euler-Lagrange equations compares to the complexity of solving N2?

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. Goes for rigor over clarity. Woodhouse has a confusing habit of doing every derivation in multiple coordinate systems at the same time.
• Kibble and Berkshire, Classical Mechanics
• Morin, Introduction to Classical Mechanics ch 6 ✔ Lives up to Morin’s usual standards. Very illuminating worked examples.
• Feynman Lectures, lecture II.19 on the Principle of Least Action
• Tong, “The Lagrangian Formalism”

Applications

• Denny, “A uniform explanation of all falling chain phenomena” in AJP.

Last updated 2 November 2024