- #1

- 4

- 0

## Homework Statement

The Lagrange method does work for some velocity dependent Lagrangian. A very important

case is a charged particle moving in a magnetic field. The magnetic field can be represented as a "curl" of a vector potential ∇B = ∇xA . A uniform magnetic field B0 corresponds to a

vector potential ∇A = 1/2 B0 x r.

(a) Check that B0 = ∇xA

(b) From the Lagrangian

[itex]\frac{1}{2}mv^{2}+e\overline{v}.\overline{A}[/itex]

show that the EOM derived are identical to the classical Newton's law with the Lorentz

force F = ev x B .

## Homework Equations

Euler-Lagrange equations:

[itex]\frac{d}{dt}(\frac{\partial L}{\partial \dot{s_{j}}})[/itex]=[itex]\frac{\partial L}{\partial s_{j}}[/itex]

Triple product:

a.(b x c)=b.(c x a)

a x (b x c)=(a.c)b - (a.b)c

## The Attempt at a Solution

For a, I have tried using the triple product:

∇ x A = 1/2 ∇x(Bo x r) = 1/2 [Bo(∇.r)-(∇.Bo)r]

Since Bo is uniform, its divergence is zero and so:

∇ x A =1/2 Bo(∇.r)

From here, I guess ∇.r=2 for the proof to work, but I really don't see it.

For part b, I am not even sure where to start it. As in how can I apply the E-L equations to it? What scares me the most is how exactly do I take the derivatives on a Lagrangian involving vectors.

Any help is welcome, thank you!