Motion in the Projected Parabola
OK, so building off my last post, the question I want to work through is, “How do free particles move in spaces that are non-euclidean?”
Let’s use Andy’s suggestion of an object sliding around on a parabola, and we’ll project that parabola onto the x-axis, to see what motion is like on the 1D projected parabolic space. But let’s take away gravity, because we want to consider free particles that are subject to no ‘external’ forces.
We know of course that in the 2D world, the particle would move with constant speed from one side of the parabola to the other. The reason it would move with constant speed is because, without gravity, there is no way for the object to change its kinetic energy, or direction for that matter.
What I want you to imagine is you are looking down into the parabola from above, and you can only see what’s happening in the x-direction. It’s like you have no depth perception. To you, the ball would seem to speed up as the ball goes down the parabola and slow down as it goes back up the parabola. That’s because some of the speed is “hidden” in the y-direction, which you can’t see. Keep this in mind, because our mathematics has to jive with this intuition.
Getting the Projected Parabolic Metric
So let’s take the parabola y = 1/2 x²
As I said in my previous post we typically measure distances like this
ds² = dx² + dy²
But we can transform this 2D euclidean space to our 1D non-euclidean space by using the fact that
dy = x dx
Plugging that into ds² = dx² + dy² gives
ds² = (1+ x²) dx²
Recognize that the term “x” is just the value of the slope of parabola at the point x, so it’s completely consistent with what I’ve talked about in the first post.
Newton’s 1st Law in the Projected Parabola
Now, I’ve already established that the object moves with constant speed in the 2D euclidean world. So let’s set a value for that velocity and just call it “v”.
With a velocity v, the distance in the 2D geometry would be s = v t or ds = v dt, where s is measure along the arc length of the parabola.
This allows us to write
v² dt² = (1+ x²) dx²
which I’ll rearrange to be
(dx/dt)² = v² / (1+ x²)
dx/dt = v / Sqrt (1+x²)
Here is where we should check out intuition, because the value dx/dt is the velocity in the x-direction. I said that the ball appear faster near x =0, because there none of the speed is hidden in the y-direction. Plugging in x =0, we get dx/dt = v. That’s good, because that’s when none of the velocity is hidden. If we take the limit as x gets really large, the velocity goes to zero. And that also makes sense because the parabola gets so steep that you can’t see any of the velocity–it’s all hidden in the y-direction.
OK, what we might want to do is write Newton’s 1st laws for a 1D parabolic geometry. So what we want is to rewrite this expression so that it looks like
d²x/dt² + Correction[x, dx/dt] = 0
And we can do this by taking a time derivative of our equation for dx/dt and rearranging.
d²x/dt² – v * d/dt [ 1/ Sqrt (1+x(t)²)] = 0
d²x/dt² + v * x / (1+x²)3/2 *dx/dt = 0
Now this is pretty ugly, but it puts it in a form like Newton’s 1st law. Our corrections term is the messy part, because it is non-linear function of x and also depends on dx/dt. It might be tempting to think of this whole term as velocity and position dependent “force”, but we don’t have to. I want to stress that this is just how particles move in our 1D projected parabolic space without forces. I’ll show you later how we can add forces.
The truth is, it would have been prettier to not take that derivative, and cast this in term of energy. One way of doing this is to multiply our expression for (dx/dt)² by 1/2 m, to get
1/2 m (dx/dt)² = KE / (1+ x²)
where KE is just a constant.
Adding Back in Gravity (the easy way)
To add back in gravity, we just have to realize that KE is no longer a constant in the 2D Euclidean space. But since the total energy in 2D euclidean space is still constant, we can write
KE = Total Energy – PE = TE – mgy = TE – mg (1/2 x²)
1/2 m (dx/dt)² = (TE – mgy) / (1+ x²)
1/2 m (dx/dt)² = (TE – 1/2 mg x²) / (1+ x²)
where TE is just a constant expressing the total energy in the 2D Euclidean space.
I’m pretty sure this works. I suppose we’ll see.
Adding Back in Gravity (the hard way)
This will be the subject of another post, but basically we’ll start with our Modified Newton’s 1st Law expression, and we’ll have to figure out how to take a gradient in a non-euclidean space.
As an aside, I’d like to point out that SO far, we haven’t had to use calculus of variations or Langrangians.