Classical Dynamics 2/II/15B 2008

[Tangent87]

Hi, I'm doing this Classical Dynamics section II question which can be found here (http://www.maths.cam.ac.uk/undergrad/pastpapers/2008/Part_2/list_II.pdf ) on page 27.

I have done most of the question but am unsure about the last part. Specifically using Hamilton's equations to show there's circular motion of radius r with the angular frequency given. I am basically just unsure of what you actually have to show, I mean they've already told us that $$r^2=x_1^2+x_2^2$$ so what actually constitutes circular motion? I can show from Hamilton's equations that

$$\stackrel{..}{x_i}=(p_3-\frac{eF}{c})\frac{e}{m^{2}cr}\frac{dF}{dr}x_i$$


for i=1,2 getting someway towards the expression for the angular frequency but don't really know where to go from here seeing as r depends on BOTH $$x_1$$ and $$x_2$$ so that differential equation isn't of SHM form!

 

[Tangent87]

I'm still stuck on this, I know that two SHM systems acting perpendicular to each other produce circular motion but that's not what we have in this case (or is it?) since the r depends on the x_i!

 

[zzzoak]

We have

$$x_i''=-\Omega^2(r) \; x_i$$

Making a square and adding

$$x_1''^2+x_2''^2=\Omega^4(r) \; (x^2_1+x^2_2)=\Omega^4(r) \; r^2.$$

RHS depends only on r, it follows and LHS too

$$a^2(r)=\Omega^4(r) \; r^2.$$

And we get

$$a(r)=-\Omega^2(r) \; r.$$

 

[Tangent87]

I see what you've done, thanks.

 

[paranormal]

Hi there! First of all, it's great that you have already made progress on this question and have a good understanding of Hamilton's equations. Let's take a closer look at the problem and see if we can clarify the concept of circular motion in this context.

In classical dynamics, circular motion is defined as motion in a circular path at a constant speed. In this question, we are dealing with a charged particle moving in a circular path under the influence of an electric field. This means that the particle is experiencing a force that is always perpendicular to its velocity, causing it to move in a circular path.

Now, let's look at the expression you have derived using Hamilton's equations:

\stackrel{..}{x_i}=(p_3-\frac{eF}{c})\frac{e}{m^{2}cr}\frac{dF}{dr}x_i

This equation describes the acceleration of the particle in terms of its position and momentum. In order to show that there is circular motion, we need to show that the acceleration is always perpendicular to the velocity, which is a characteristic of circular motion.

To do this, we can rewrite the equation in terms of the radial position and momentum, denoted by r and p_r respectively. This can be done by using the relation r^2=x_1^2+x_2^2, as given in the question. Once we have the equation in terms of r and p_r, we can use the definition of circular motion to show that the acceleration is always perpendicular to the velocity.

I hope this helps clarify the concept of circular motion in this context and gives you a starting point for solving the problem. Good luck!