Runge-Lenz vector with perturbation potential

In summary, the conversation discusses the time derivative of the Lenz vector in the case of a potential ##\sim 1/r## and the integration of the perturbation potential/force into the definition of the Lenz vector. The speaker has proven that the time derivative of the Lenz vector is zero, but is unsure of how to integrate the perturbation potential/force into the definition. The other person suggests that this may be due to approaching the question incorrectly and asks for clarification on the mathematical proof.
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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?
 
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  • #2
breadlover98 said:
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.)

breadlover98 said:
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.
 

1. What is the Runge-Lenz vector with perturbation potential?

The Runge-Lenz vector with perturbation potential is a mathematical concept used in quantum mechanics to describe the motion of a particle in a central potential with a small perturbation. It is derived from the classical Runge-Lenz vector, which describes the conservation of angular momentum in a central force field.

2. How is the Runge-Lenz vector with perturbation potential calculated?

The Runge-Lenz vector with perturbation potential is calculated using the Hamiltonian operator, which is a mathematical representation of the total energy of a system. It is derived from the classical Runge-Lenz vector by applying quantum mechanical principles.

3. What is the significance of the Runge-Lenz vector with perturbation potential?

The Runge-Lenz vector with perturbation potential is significant because it provides a way to calculate the energy levels and wavefunctions of a particle in a central potential with a small perturbation. It also helps to understand the effects of perturbations on the motion of a particle in a central potential.

4. How does the Runge-Lenz vector with perturbation potential relate to the Kepler problem?

The Runge-Lenz vector with perturbation potential is closely related to the Kepler problem, which describes the motion of a particle in a central force field. The classical Runge-Lenz vector was first introduced to solve the Kepler problem, and the quantum mechanical version is used to solve the quantum Kepler problem.

5. What are some applications of the Runge-Lenz vector with perturbation potential?

The Runge-Lenz vector with perturbation potential has many applications in quantum mechanics, including in the study of atomic and molecular systems. It is also used in the calculation of energy levels and wavefunctions for particles in a central potential with a small perturbation, and in understanding the effects of perturbations on the motion of these particles.

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