Noether's theorem and constructing conserved quantities

[CAF123]

Homework Statement

A particle of mass m and charge e moving in a constant magnetic field B which points in the z-direction has Lagrangian ##L = (1/2) m( \dot{x}^2 + \dot{y}^2 + \dot{z}^2 ) + (eB/2c)(x\dot{y} − y\dot{x}). ##

Show that the system is invariant under spatial displacement (in any direction) and find the associated constants of the motion.

Show that the system is invariant under an infinitesimal rotation about the z-axis and find the associated constant of the motion.

Homework Equations

[/B]
arbritary spatial displacement, ##\mathbf r' = \mathbf r + \mathbf a ##

infinitesimal rotation, ##\mathbf r' = \mathbf r + (\hat n \times \mathbf r) \delta \theta##

Noether's theorem : $$\frac{\partial L}{\partial \dot{q_i}} w_i + G = \text{const}$$

The Attempt at a Solution

[/B]
I can write ##\mathbf r' = (x+a_1) \hat x + (y+ a_2)\hat y + (z+a_3)\hat z = x' \hat x + y' \hat y + z' \hat z## then read off the coordinates after the shift. Plug these into lagrangian and show it is invariant up to a total derivative ##\dot{F}## where ##F = F(x,y) = -(eB/2c) (-a_1 y + a_2 x)## and hence e.o.ms don't change and so system invariant. Does this seem fine? I think there will be three constants of motion here. Use Noether's theorem to find them, (as per the question), so $$I_1 = \frac{\partial L}{\partial \dot{x}} \cdot 1 + G,$$ where ##G = -\epsilon F##. What is epsilon in this case? I was thinking I could write ##\mathbf a = \epsilon_1 \hat x + \epsilon_2 \hat y + \epsilon_3 \hat z## where all the ##w_i##'s in Noethers theorem are set to one here and ##\epsilon_i = a_i##. So that ##I_1## becomes $$\frac{\partial L}{\partial \dot{x}} \cdot 1 + \epsilon_1 \frac{eB}{2c} (-\epsilon_1 y + \epsilon_2 x)$$ Is it fine? Thanks!

 

[BigJDubs]

For the second part, I am more confused. What is ##w_i## and ##\epsilon## here? Is it the same as before? Is it just ##\epsilon \hat n \times \mathbf r##? Any help would be appreciated.