Classical mechanics: Central potential, trajectory

[fluidistic]

Homework Statement

Determine the possible trajectories of a particle into the following central potential: U(r)=U_0 for r< r_0 and U(r)=0 for r>r_0.

Homework Equations

Not sure. What I used: Lagrangian+Euler/Lagrange equations.

The Attempt at a Solution

I used polar coordinates but I'm not sure it's well justified. Since we're dealing with a central potential problem, isn't the motion restricted to a 2 dimensional plane? If so, then I think it's safe to use polar coordinates.
Ok so my Lagrangian is L=T-V=\frac{m(\dot \theta r^2 + \dot r ^2 )}{2}-U(r). I notice that \theta is cyclic, thus the angular momentum is conserved.
Using Lagrange's equations for \theta, I reached the differential equation \dot r r = cte. For r, I reach \ddot r - \dot \theta r =0.
I know I made an error, I reach the same motion equation regardless what U(r) thus r is.
Hmm... Since \theta is cyclic, does this imply that \dot \theta =0?
If so, then \ddot r =0. But this is senseless since it would imply that \dot r is also a constant, and thus that r is also a constant from my first use of Lagrange equations...
It might be a little too late to do physics for me now, but I'm willing to put the effort.
Any help will be appreciated. Thanks a lot!

Edit: I notice an error for the 1st Euler/Lagrange equation conclusion. I do reach that r is constant instead of r \dot r, which still doesn't make sense to me.

 

[hikaru1221]

There is a silly mistake at T (kinetic energy). Check that
(by the way, d\theta/dt=0 is meaningless. That would mean the particle's motion, including the initial state, is restricted to radial direction)

 

[fluidistic]

There is a silly mistake at T (kinetic energy). Check that
(by the way, d\theta/dt=0 is meaningless. That would mean the particle's motion, including the initial state, is restricted to radial direction)

Thanks for pointing this out. Indeed.
T is in fact \frac{m(\dot \theta ^2 r^2 + \dot r ^2 )}{2}.
The equations of motion I reach are now 2r \dot r \dot \theta + \ddot \theta r^2=0 and \ddot r - \dot \theta ^2 r =0.
I also know that r^2 \theta = constant.
I don't know if I can divide any of these equations by r. As far as I know, r could be worth 0 and thus I can't divide by r.
I don't know how to proceed in order to reach the trajectories; or in other words, how to solve the motion equations.
Also I notice that there's no difference at all for r>r_0 or r<r_0 in my equations. It seems wrong to me.

 

[George Jones]

The \partial L / \partial r term involves \partial U / \partial r = d U / dr. What is d U / dr? It isn't zero everywhere.

 

[fluidistic]

The \partial L / \partial r term involves \partial U / \partial r = d U / dr. What is d U / dr? It isn't zero everywhere.

Ah yes you're right. It's worth 0 everywhere except in r=r_0 where it's worth \pm \infty depending upon if U_0 is greater or lesser than 0.
So U(r) isn't analytic. How do I deal with this problem?!

 

[George Jones]

Are you familiar with the Dirac delta "function"?

 

[fluidistic]

Are you familiar with the Dirac delta "function"?

A bit. I've seen some relations and the formal definition of it (if I remember well it was the limit of a sequence) but I'm not so comfortable with it.
Out my memory, I remember that \int g(x) \delta (x-a) dx=g(a) where the integral bounds cover of course an interval containing x=a.
I didn't check in wikipedia so my memory might fail. Also a friend of mine once told me it was a double abuse of notation to write the integral I just wrote, but I didn't really understood why.
So in my example, a=r_0. And g(x) would be U(r). So basically... err no that can't be... Hmm.

 

[George Jones]

Have you seen the connection between the Heavisde step function and the Dirac delta function?

With experience, it is actually possible to see the answer to the original problem without doing any calculations (somewhat similar to billiard ball collision problems). We should go through the calculations, though.

 

[fluidistic]

Have you seen the connection between the Heavisde step function and the Dirac delta function?

With experience, it is actually possible to see the answer to the original problem without doing any calculations (somewhat similar to billiard ball collision problems). We should go through the calculations, though.

About the Heaviside function: Heard of it before but never checked it in details. I just did in wikipedia and its connection to the Dirac Delta function seems to be that H(x)=\int _{-\infty} ^x \delta (t)dt. So in my case it'd be worth 0 for x&lt;r_0 and 1 for x&gt;r_0. But I don't see how this can help. I've checked in Golstein's book (1st edition) on Classical dynamics and to my surprise there's a short comment on Dirac's Delta and nothing on Heaviside function.
So the problem seems simple but it requires the use of Dirac's delta due to the singularity of the potential function. I'll help for further help here as I'm in unknown territory (I'll still try to figure out things if I can). And thank you so far.

 

[George Jones]

H(x)=\int _{-\infty} ^x \delta (t)dt

Can you invert this?

So in my case it'd be worth 0 for x&lt;r_0 and 1 for x&gt;r_0.

Consequently, U \left( r \right) = U_0 H \left( r_0 - r \right). What does this give for dU / dr?

But I don't see how this can help. I've checked in Golstein's book (1st edition) on Classical dynamics and to my surprise there's a short comment on Dirac's Delta and nothing on Heaviside function.
So the problem seems simple but it requires the use of Dirac's delta due to the singularity of the potential function. I'll help for further help here as I'm in unknown territory (I'll still try to figure out things if I can). And thank you so far.

Look up "impulsive force" in Goldstein or your text or ...

 

[fluidistic]

Can you invert this?

Do you mean that I find the inverse of the Heaviside function?

Consequently, U \left( r \right) = U_0 H \left( r_0 - r \right). What does this give for dU / dr?



Look up "impulsive force" in Goldstein or your text or ...

The first edition of Golstein as only 1 exercise about impulsive forces. It seems rather tough to me and doesn't seem to involve any special function.
Ok back to your expression. Now I understand that U(r)=U_0 H(r_0-r).
I also understand I must find \frac{d U}{dr}. If I'm not wrong, I think this is simply worth U_0 \delta (r_0 -r).
So as I had, \frac{d U}{dr}=0 for r\neq r_0 which makes sense.
Now in r=r_0 it's worth \infty as it should I guess.
How would I continue? I'm not seeing where we're going.

 

[hikaru1221]

That expression U(r)=U_0H(r_0-r) along with one ODE and the expression r^2\dot{\theta}=C=const will give you an ODE of r. As you want to stay cautious with dividing by r, you may simply consider ODE at r \neq 0. You will get something like this:
f(\ddot{r},\dot{r},r)=\delta(r-r_0)

Here's the trick: observe that the change only happens at r=r_0 (this is what George meant by impulsive force: an immediate change without change in position). So deal with the ODE within the interval: (r_0^-;r_0^+). Notice that: \int^{0^+}_{0^-}\delta(x)dx=1

 

[fluidistic]

Thanks guys for all the help so far.
To hikaru, could you be a little more specific about the ODE's I should get? I've redone all the algebra. I do reach \dot \theta r^2 = constant. I do reach \ddot \theta r ^2 + 2 \dot \theta \dot r r =0. Is this the first ODE you talk about?
I don't see how I can "mix them" along with U(r)=U_0H(r_0-r) in order to get a single ODE depending on r.
And ok for your trick, I'll use it.

Edit: If \frac{dU}{dr}=U_0 \delta (r-r_0 ) then I reach the ODE \dot \theta ^2 r + U_0 \delta (r-r_0)-\ddot r =0. I have no idea if this is right.

 

[George Jones]

\dot \theta ^2 r + U_0 \delta (r-r_0)-\ddot r =0. I have no idea i this is right.

Good! Are there some missing m's?

Now use your other equation to eliminate \dot \theta.

 

[fluidistic]

Good! Are there some missing m's?

Now use your other equation to eliminate \dot \theta.

Thanks for helping me. :)
Yes I miss one m (my draft hasn't this error so I suppose it's a typo error). Edit: OH actually 2 I see.
This should be m \dot \theta ^2 r + U_0 \delta (r-r_0)-m \ddot r =0(*).
Now I'm not sure what equation to use. I assume you mean \dot \theta r^2=K_1 but I don't see how I can use it in order to give rid of \dot \theta in (*).

Edit2: My bad nevermind, I think I know how!

 

[fluidistic]

I get \frac{mK_1 ^2}{r^3}+U_0 \delta (r-r_0)-m \ddot r =0. This is true for r\neq 0.
I need to solve this equation for r... Ok I'll think about how to do but I'm not that confident.
Edit: Ok it doesn't look that bad. I still have a doubt about the limits of some integrals (funnily I've helped someone on this forum yesterday night about the limits of an integral...); let me show you.
(*) is equivalent to \ddot r = \frac{K_1 ^2}{r^3} + \frac{U_0 \delta (r-r_0)}{m}.
I integrate with respect to time in order to get \dot r (with the idea of doing it again to get r).
So I have \dot r=K_1 ^2 \int \frac{dt}{r^3} + \frac{U_0}{m} \int \delta (r-r_0)dt. My guess is that the limits of the integrals are -\infty and +\infty.
I'd like a confirmation if I'm on the right track and the limits are correct.

 

[George Jones]

Maybe I shouldn't give this hint, but ... reduction of order.

 

[fluidistic]

Maybe I shouldn't give this hint, but ... reduction of order.

Oops I didn't see this message. I edited my above post and was looking for a different approach. Could this work too?

Edit: If reduction of order is the only "easy" way to go then I'm already lost on what is the characteristic polynomial.

 

[George Jones]

So I have \dot r=K_1 ^2 \int \frac{dt}{r^3} + \frac{U_0}{m} \int \delta (r-r_0)dt. My guess is that the limits of the integrals are -\infty and +\infty.
I'd like a confirmation if I'm on the right track and the limits are correct.

This equation is correct, but since r \left( t \right) is to be determined, we don't know how to do the integrals.

If reduction of order is the only "easy" way to go then I'm already lost on what is the characteristic polynomial.

See Type 2 (example almost halfway down the page) from

http://www.cliffsnotes.com/study_guide/Reduction-of-Order.topicArticleId-19736,articleId-19724.html .