Show that the Laplace-Runge-Lenz vector is conserved using poisson brackets

Question

(I realise similar Phys.SE questions already exist but there is no answer with the Poisson bracket notation, I'll take this down if someone lets me know I should have commented in the existing question.)

I am trying to show that the Poisson bracket between the Hamiltonian and the Laplace-Runge-Lenz vector vanishes, i.e.

$$\left\{H,A\right\}_{PB}=0$$

where $\vec{A} = \left(p \times L\right) - m k\cdot \hat{r}$, and the Hamiltonian is for an orbit is given by, $$H = \frac{m}{2}\left(\frac{dr}{dt}\right)^2 + \frac{k}{r}$$

I have been trying to use tensor notation to write out the cross product term and the fundamental Poisson brackets but am not having any luck.

Answer 1

Recall $A^2 = m^2k^2+2mEL^2$. We can first prove the magnutude is constant:

$$[A^2,H] = [m^2k^2+2mEL^2,H] = [2mEL^2,H]$$

$$ [A^2,H] = 4mEL[L,H] = 4mEL \dot{L} $$

Since $\dot{L} = 0 $, this implies $\dot{A} = 0$. Not we jus need to show the components are constant. Notice:

$$\vec{A} = <yL_z-mk,xL_z-mk,0>$$

Thus $A_x = A_x(y)$ and $A_y=A_y(x)$. Since $A_x^2+A_y^2= constant$ and each are functions of different variables, each must be constant.