00:01
Okay, we're supposed to write down a lagrangian for this system.
00:05
We have a mass big m that slides back and forth on a frictionless rail, and then a mass little m that's hanging from that and swinging back and forth like a pendulum, okay? and we're supposed to write out a lagrangian for this.
00:20
The most important thing to remember about writing out a lagrangian is that when you figure out the kinetic energy term that you have to relate it to an inertial reference frame.
00:41
And because this mass little m is attached to a mass big m that's sliding back and forth, the mass big m is not an inertial frame.
00:52
It's non -inertial because it's accelerating back and forth.
00:57
So you can't just write out, it's not that trivial to just write out the kinetic energy.
01:06
You just have to be careful when you do that.
01:09
So i'm gonna define these two other variables called x1 and y1.
01:15
Y1 is okay, non -inertially, because it only depends on angle.
01:20
It doesn't depend on x.
01:24
So y1 dot looks like that.
01:27
X1 dot looks like that, okay? and then our kinetic energy term.
01:37
So little x is fine as an inertial reference frame.
01:44
So we're thinking about the inertial value when we're thinking about the kinetic energy of big m.
01:54
We have to use x1 and y1 when we're thinking about little m and we have to write them in terms of our actual generalized coordinates, x and theta, okay? and then we can expand that.
02:27
And then our potential energy just has to do with how high the spring is.
02:33
And i'm gonna specify my zero gravitational energy up here at the wire where our big mass is sliding back and forth.
02:41
The gravitational potential only affects the hanging mass and not the sliding.
02:48
And that's what it looks like because it's really just mgh with a minus sign because i'm putting my zero up at the wire so it's below that mgl cos theta.
03:02
So my lagrangian looks like this.
03:20
Okay, so there's the lagrangian.
03:22
So now let's figure out the equations of motion.
03:25
So first the x equation, we need the derivative of l with respect to x dot, okay? and partial of l with respect to x is zero.
03:48
And so we get an equation of motion where we take the time derivative of that.
03:52
I'm not actually gonna evaluate that time derivative.
03:57
I'm just gonna write it out like this because here's the thing, because it equals zero, there's no potential term in x.
04:12
So the time derivative of what's inside there is zero.
04:18
So this thing, big m x dot plus little m x dot plus ml cosine theta theta dot, that's a constant of the motion.
04:28
And we'll be able to exploit that later on.
04:33
For the theta equation, so that was the x equation.
04:36
For the theta equation, partial l partial theta dot, partial l with respect to theta.
04:49
Now it does depend on angle, but only in the potential energy.
04:54
Well, it's got an angle dependence in the kinetic energy because of the two l cos theta x dot theta dot term.
05:04
And it has another angle dependence in the potential energy term.
05:19
And so this equation, again, i'm not gonna take that time derivative.
05:25
They'll just be messy.
05:26
And i'm gonna wanna simplify the equation first before i do that.
05:32
Since we're mostly interested in small oscillations.
05:39
So there's my second equation.
05:42
And like i said, i can expand this and take those derivatives if i want to, but there's no real reason to do it.
05:55
So for the next part, for small oscillations, so theta is small, x is small.
06:13
So really it's not x that's small, it's x dot that's small...