Proving Runge-Lenz Vector Time Derivative is Constant

• the_kid

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?

Any help on this?

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

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

1. What is the Runge-Lenz vector and why is it important?

The Runge-Lenz vector is a conserved quantity in classical mechanics that describes the orientation and magnitude of the orbital angular momentum of a particle in a central force field. It is important because it allows us to solve the otherwise difficult problem of determining the orbit of a particle under a central force.

2. Why is it necessary to prove that the time derivative of the Runge-Lenz vector is constant?

Proving that the time derivative of the Runge-Lenz vector is constant is necessary because it confirms that the vector is truly conserved and does not change over time. This is a fundamental property of the vector and allows us to make accurate predictions about the motion of a particle under a central force.

3. How is the time derivative of the Runge-Lenz vector derived?

The time derivative of the Runge-Lenz vector, also known as the Laplace-Runge-Lenz vector, can be derived using the equations of motion for a particle under a central force. By taking the time derivative of the vector and manipulating the resulting equations, we can show that the derivative is indeed constant.

4. What implications does the constant time derivative of the Runge-Lenz vector have on the motion of a particle?

The constant time derivative of the Runge-Lenz vector implies that the particle's orbit will remain unchanged over time, and its angular momentum will remain constant. Additionally, it allows us to determine the shape and orientation of the orbit, as well as the magnitude of the orbital angular momentum, without needing to solve for the particle's position at each point in time.

5. Are there any real-life applications of the Runge-Lenz vector and its time derivative?

Yes, the Runge-Lenz vector and its time derivative have practical applications in various fields, including celestial mechanics, astrophysics, and spacecraft trajectory planning. They are also used in quantum mechanics to describe the motion of particles in a Coulomb potential, such as an electron orbiting a nucleus in an atom.