1. Andrew Conti says:

Jennifer Coopersmith you are amazing. I just finished reading, cover to cover, your book “The Lazy Universe”. It is simply awesome. I so wished you were there teaching me as a freshman in college. I am actually a chemist but my physics background was not great and I had to play catchup for years. Your simple and clear explanation of physics is extroardinary. Now I get it. Thanks.

2. charliebrownian says:

Thank you for writing “The Lazy Universe!” I’ve been working through it and it’s been very helpful in laying out the derivation of Hamilton’s Principle (the Principle of Least Action), so I can see now why systems try to minimize the time-integral of the Lagrangian. I’ve also been through a few proofs of where the Euler-Lagrange equation comes from, how it can be applied to Hamilton’s Principle to obtain the equations of motion for various coordinates, and how the Euler-Lagrange equation is coordinate-invariant so that as long as we can express one set of coordinates as functions of some other set of coordinates (and possibly time), Lagrange’s equations of motion still hold.

The thing that’s tripping me up is that q(t) in the Euler-Lagrange equation doesn’t have to be a spatial coordinate, but can be just about anything (“changes in capacitance, surface tension, magnetic field, phase of a wave, strain in a beam, pressure within a fluid, and so on” on pg. 72). I’ve been through the example on damped resonant circuits you have in the appendix and can see that Lagrange’s equations of motion work when q(t) represents the charge in a wire (along with other examples in electrodynamics: Feynman shows how it can be used to get the Lorentz force equation and Poisson’s equation, for example), so I can see that it *does* work for q(t)’s other than spatial coordinates, but I don’t understand *why.* Is there a proof somewhere of why q(t) is allowed to be so many different physical quantities? How did it ever occur to someone to try to use Lagrange’s equations to solve for something other than a spatial coordinate?

1. jennifercoopersmith says:

Hi Charlie, thanks for your appreciation, and welcome to the slowly growing band of people who begin to realize that the Principle of Least Action underlies all physics. Happy to answer your questions (in brief) but please direct them to me at j.coopersmith@latrobe.edu.au