Lagrangian for a particle moving in x-y plane in a constant B-field

[spacetimedude]

Homework Statement

Not sure if the link is showing. But it's imgur.com/a/LEvd0

Homework Equations

The steps I've taken so far as written in the attempt section below is correct.

The solution provided then proceeds with letting $z = x + iy$ and setting $\ddot z+i \omega \dot z = 0$. Then $z(t)=X\exp(-i\omega t) + Y$ where X, Y are complex.
Lastly, $X=Aexp{(i\omega t)} * \text{something illegible}$ and $Y=D+iE$ and then says this leads to the x and y equations shown in the question.

The Attempt at a Solution

We first solve the lagrangian equation by splitting x and y terms. So for x:
$$\frac{d}{dt} \frac{\partial L}{\partial \dot{x} } - \frac{\partial L}{\partial x} = 0$$
and $$\ddot x -\omega \dot y =0.$$
Similarly for y:
$$\ddot y +\omega \dot x= 0$$

I'm not quite sure what to do after this step. As explained in the previous section, the solution sets $z=x+iy$ but the steps afterwards are very unclear and hard to read.

Any help will be appreciated.

 

[Chandra Prayaga]

Step 1: Set z = x + iy, use the two differential equations you got for x and y, to get the differential equation for z. You can proceed from there.

 

[spacetimedude]

Step 1: Set z = x + iy, use the two differential equations you got for x and y, to get the differential equation for z. You can proceed from there.

That's the part where I'm confused. How would I incorporate the x and y differential equations to z=x+iy?

 

[Chandra Prayaga]

Can you multiply one of the two equations with i and then add the two equations? Will that help?

 

[spacetimedude]

Can you multiply one of the two equations with i and then add the two equations? Will that help?

Thank you.

Multiply i to the y DE and add the two equations so:
$$ \ddot x - \omega \dot y + i\ddot y +i\omega \dot x =0. $$
Rearrange and it becomes
$$\ddot z +i\omega \dot z = 0.$$
Does this then imply
$$\dot z + i\omega z = Y$$ where Y is complex?
Then solving for z leads to $$z = X\exp(i\omega t) +Y$$ where X and Y are complex (the Y here is different by a factor from Y before but still a complex).

Now, another part I am confused is, the solution says Y can always be written as Y = D+iE which I understand since it's a complex, but it also says X can be written as something like $A\exp(iωt_a)+C\exp(iωt_b)$ but I can't really read off what it says. To conform with the solutions of x and y given, what could this X be?

 

[Chandra Prayaga]

Use the boundary conditions. z = x + iy, and both x and y are specified for two values of t. That will give you the value of z at those times. Compare that with the solution that you got above for z. That should give you X nd Y