Runge-Lenz vector with perturbation potential

[breadlover98]

Homework Statement

Consider the Kepler problem

$$m \ddot{\vec{r}} = -\alpha \frac{\vec{r}}{r^3}, \quad \alpha = GMm$$

Another conserved quantity, called the Runge-Lenz vector, is given by

$$\vec{F}_L = \vec{p} \times \vec{L} - m \alpha \frac{\vec{r}}{r}$$

Now imagine the gravitational force is perturbed by another central force

$$\vec{F}' = f(r) \frac{\vec{r}}{r}$$

where $f(r) \sim 1/r^3$. As a result of this, the Lenz vector is not conserved anymore. Hence, find:

$$\frac{\mathrm{d}\vec{F}_L}{\mathrm{d}{t}} = \dot{\vec{F}}_L$$

and discuss the effect of this perturbation on the motion.

Relevant Equations

The given equations are included in the homework statement.

For the case that there is only a potential $\sim 1/r$, I have already proven that the time derivative of the Lenz vector is zero. However, I'm not sure how I would "integrate" this perturbation potential/force into the definition of the Lenz vector (as it is directly defined in terms of the gravitational potential). Is there a more general/extended version of this definition or am I approaching this question wrong?

 

[strangerep]

For the case that there is only a potential $\sim 1/r$, I have already proven that the time derivative of the Lenz vector is zero.

Well, how did you prove that $\,dF_L/dt = 0\,$ ? (I.e., sketch out the math for us.)

However, I'm not sure how I would "integrate" this perturbation potential/force into the definition of the Lenz vector (as it is directly defined in terms of the gravitational potential). Is there a more general/extended version of this definition or am I approaching this question wrong?

I suspect you're "approaching it wrong". Hopefully, after you answer my question above, this should become clearer.