Proving Runge-Lenz Vector Time Derivative is Constant

[the_kid]

I'm trying to prove that the time derivative of the Runge-Lenz vector is constant. Any ideas on how I would go about doing this?

 

[the_kid]

Any help on this?

 

[vanhees71]

Simply take the time derivative! The definition of the Lenz vector is

\vec{A}=\vec{p} \times \vec{L}-m \alpha \frac{\vec{r}}{r}.

To show that this is conserved for the potential V(r)=-\alpha/r, we note that the angular momentum is conserved, and we have

\vec{L}=m \vec{r} \times \dot{\vec{r}}=m r^2 \vec{\omega}=\text{const},

where \vec{\omega} is the momentary angular velocity.

Further we have

\frac{\mathrm{d}}{\mathrm{d} t} \frac{\vec{r}}{r}=\frac{\dot{\vec{r}}}{r}-\frac{\dot{r} \vec{r}}{r^2} = \vec{\omega} \times \frac{\vec{r}}{r}

and

\dot{\vec{p}}=-\vec{\nabla} V(r)=-\frac{\alpha}{r^3} \vec{r}

and thus

\dot{\vec{A}}=\dot{\vec{p}} \times \vec{L}-m \alpha \vec{\omega} \times \frac{\vec{r}}{r}=-\frac{\alpha}{r^3} \vec{r} \times m r^2 \vec{\omega}-m \alpha \vec{\omega} \times \frac{\vec{r}}{r}=0.