3 ms·

I think it's an even better story how he "guessed wrong" with Hamiltonian mechanics, also; the only difference is that he never fixed the mistake and it wasn't

I think it's an even better story how he "guessed wrong" with Hamiltonian mechanics, also; the only difference is that he never fixed the mistake and it wasn't really fixed until the early 20th century with the development of optimal control. See "The Brachistochrone Problem and Modern Control Theory" (http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.26.825&rep=rep1&type=pdf http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.26....) The TL;DR is that classical mechanics can be formulated as finding a time history of position and velocity which minimizes a certain function of them both. Hamiltonian mechanics essentially optimizes the function over position and velocity as if position and velocity were independent, then adds the constraint that velocity be the time derivative of position via a Lagrange multiplier. This Lagrange multiplier is the momentum. Of course, no one in a physics course actually learns it that way; undergraduates just memorize some equations and graduate students, if they are lucky, learn if from the Hamilton-Jacobi equation (which is how Hamilton developed it in the first place) where the momenta become the spatial gradient of the function to be minimized when evaluated at all positions. (It's exactly the same as the Bellman return function in dynamic programming; hence the name "Hamilton-Jacobi-Bellman equation.") IMHO, neither of these other developments is as intuitive as the Lagrange multipler one; but because Hamilton's original formulation obscures it, no one in physics learns it that way. A fun postscript: when you consider the equations of classical mechanics in terms of position and momenta (this combination is what physicists call "phase space") they form a manifold with a special property called "symplectic"; symplectic geometry can be formulated in terms of matrices over the field of quaternions. According to some people, this idea was a big breakthrough, but I don't believe Hamilton himself, despite having formulated both, ever noticed.

starwed 16y ago
"IMHO, neither of these other developments is as intuitive as the Lagrange multipler one; but because Hamilton's original formulation obscures it, no one in physics learns it that way." I'm pretty sure learning a bit about Lagrange multipliers is standard for a grad classical mechanics class. Granted, I've forgotten all I learned back then, b/c it has never come up in anything else I do, but it was definitely part of the course.
- ArbitraryLimits 16y ago
  I believe it used to be standard to use Lagrange multipliers to handle constraints on the state space in R^n as an alternative to using generalized coordinates. For example, when you handle the case of motion on the surface of a sphere you can either use R^3 with a Lagrange multiplier enforcing the constraint that you stay on the surface of the sphere, or you could use latitude/longitude as coordinates. The interpretation of the momenta themselves as Lagrange multipliers is completely nonstandard, though. Try googling "hamiltonian mechanics lagrange multipliers"; you'll get either examples like the sphere thing or hits from optimal control tutorials.

starwed 16y ago

"IMHO, neither of these other developments is as intuitive as the Lagrange multipler one; but because Hamilton's original formulation obscures it, no one in physics learns it that way." I'm pretty sure learning a bit about Lagrange multipliers is standard for a grad classical mechanics class. Granted, I've forgotten all I learned back then, b/c it has never come up in anything else I do, but it was definitely part of the course.

ArbitraryLimits 16y ago
I believe it used to be standard to use Lagrange multipliers to handle constraints on the state space in R^n as an alternative to using generalized coordinates. For example, when you handle the case of motion on the surface of a sphere you can either use R^3 with a Lagrange multiplier enforcing the constraint that you stay on the surface of the sphere, or you could use latitude/longitude as coordinates. The interpretation of the momenta themselves as Lagrange multipliers is completely nonstandard, though. Try googling "hamiltonian mechanics lagrange multipliers"; you'll get either examples like the sphere thing or hits from optimal control tutorials.