Eli5: Please explain the Stationary-action principle
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You’re essentially trying to find the shortest between between two points on a curved surface. A simple analogy is “how do I know I’m at the top of a (smooth) hill?” If you’re at the top, short steps are just about flat, but if you’re on the side of the hill, a short step might take you either uphill or downhill. Stationary action extends this to paths between two points, where the distance you have to travel is like the height of the hill. If your path is the shortest path, small deviations (short steps) won’t affect the distance (height) very much, but if your path isn’t the shortest path, small deviations will change the distance. Mathematically, it’s easier to figure out the path that doesn’t change lengths than the shortest path. The stationary action principle says those two paths are the same, so you might as well do it the easy way.
So the Hamiltonian principle is a principle is the idea thay connects some mathematics to physics.
Here is the math: Imagine you got two points. Some path connects the two points. Now introduce some parameter that can describe a small fraction of that path. You can vary this parameter any way you want giving you some curve. You might be interested in something like: what is the minimal path leght?
So basically, you got some function for the path f. And f is a function of some variable like x. You can make a function with f and its derivitives so that some condition is satisfied. That is the Lagrange function.
And action is defined as the integral of the L function with respect to the variable of your path.
So action is a number we can assign to paths.
Lets look at how this formalism looks for the shortest path between two points:
S=integral (dl) (between the two end points). Now dl is our L function lets see what that is. dl is some part of a curve that is basically a straight bit. So we can use the angle that its tangential line makes with the x axis lets call it phi. You got a right triangle with sides dx, dy and hypotenuse dl and we got that angle phi. cos(phi)=dx/dl so dl=dx/cos(phi) = sqrt(1+tan²(phi))dx. We just rewrote 1/cos but we won. tan(phi) is the derivitive of your function f at some point so we get dl=sqrt(1+f’²(x))dx.
We can integrate with respect to x. S will be the leght of the path here.
So the question is when is this minimal? I don’t know, but we can figure out when its stationary. Now a stationary point is where your function doesn’t have first order changes, its first derivitive is 0, its slope is 0. That is either a minimum, a maximum or a saddle point. So we can introduce some kind of derivitive for L. We call that variation. Now the variation equals 0 is the equation for the stationary point. Its an equation for f. What f satisfies this equation?
Now this is quite similar to an equation of motion, where you have an equation and you are looking for a function that satisfies it. The joke is that this will be a method of putting together an equation of motion.
So you can get the stationary equation called the Euler-Lagrange equation doing a partial integral but it just ends up being this:
Take the derivitive of L(f(x),f'(x),x) with respect to f(x) and lets call it F, and lets take the derivitive of L with respect to f'(x) which gives us something we call p. The E-L equation is E=F-dp/dx. So E=dL/df – d/dx dL/df’. And the stationary (and non dissipative) case requires this to equal 0.
For mechanical problems E will be the equation of motion.
The Hamiltonian principle states that paths for stationary action will be physical paths.
The physics:
So its a fancy way of saying that for the Lagrange function of a system, the E-L equation is the equation of motion.
In physics we often have time as the variable for our functions. So L is a function of r and r’ and they are functions of time.
And L(r(t),r'(t),t)=K-V
Where K is the kinetic energy and V is the potential. If you put together L like that the resulting E-L equation will be the equation of motion. Lets look at an example.
Free fall, K=½mv² and V=mgy
L=½mv²-mgy
F=-mg
p=mv
E=-mg-mv’
-mg=ma
We got the equation of motion for a free falling mass. Great!
The principle of stationary action is what tells use which path is the physical path. We consider all possibilities, assign a number to all paths, so we get a function that takes a path as an input and gives us a number, and we are looking for the stationary points of that function.
The big thing is that this idea, Lagrange functions, the Hamiltonian principle, etc can be applied generally to many problems, from mechanics through optics to electro dynamics. Once you figure out what the action should be (quite often you only have one option for the Lagrange function) this formalism gives you back stuff like Snell’s law or the relativistic form of Maxwell’s equations.
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