Mechanics, charge in vector potential

[WarnK]

Homework Statement

A particle of charge q in a vector potential

$$\vec{A} = (B/2) (-y\vec{i} + x\vec{j} )$$

show that a classical particle in this potential will move in circles at an angular frequency $$\omega_0 = qB/ \mu c$$, $$\mu$$ is the mass of the particle.

Homework Equations

The Attempt at a Solution

I write a Hamiltionian as

$$H = \frac{1}{2\mu} \left( \vec{p} - \frac{q}{c} \vec{A} \right)^2$$

and then get

$$\mu \dot{x} = p_x + \frac{qB}{2c}y$$
$$\mu \dot{y} = p_y - \frac{qB}{2c}x$$

and then I'm lost. I'm not sure if this is right, and if it is: how do I see how this gives circular orbits and what the angular frequency is?

 

[siddharth]

Write down the Lagrangian for the particle. What happens to $$\vec{A} \cdot \vec{r}$$? So, what are the cyclic coordinates and what's conserved?

Once you figure out what the constants of motion are, you can solve the last two equations you wrote.

 

[javierR]

If you have to use the Hamiltonian formulation, don't forget the other set of equations
p'= -dH/dq.
The 4 1st order equations here correspond to the 2 2nd order equations of the Lagrangian formulation. Either way, they are coupled, and you can decouple them in a simple fashion. Circular motion corresponds to harmonic motion in each dimension, so if you can show the equations are of the form
x'' +w^2 x+constant=0
then that's it (cause you should know the solutions of such an equation already);
then you can pull off the angular frequency (speed) w. The constant appears from integration but you can verify yourself that it's zero.