Conservation of the Laplace-Runge-Lenz Vector

In summary, the conversation is about the conservation of the Laplace-Runge-Lenz vector using the Hamiltonian of a two-body system. The speaker is having trouble showing the conservation and is seeking help on how to get it from the Hamiltonian. They mention taking the square of the LRL vector to find the energy but are unsure due to uncertainty. The other person suggests using commutators to calculate it.
  • #1
fisica1988
3
0
Hmm...Latex doesn't seem to be working at the moment...

How does one show the conservation of the Laplace-Runge-Lenz (LRL) vector using the Hamiltonian of the two-body system? Showing it's conserved otherwise it's not hard. You can take the time derivative of the LRL vector and show that it's zero or a couple other ways which I worked out before (I would type it out but Latex seemingly disabled makes it tedious and cumbersome). The one thing I can't figure out is how to get the conservation of the LRL vector from the Hamiltonian of the two-body system. What I did was take the square of the LRL vector and find the energy from that but then it becomes uncertain.
 
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  • #2
You know how to calculate commutators?
 
  • #3
Ah yes yes, thank you, I got it now.
 

FAQ: Conservation of the Laplace-Runge-Lenz Vector

1. What is the Laplace-Runge-Lenz vector?

The Laplace-Runge-Lenz vector is a conserved quantity in classical mechanics that describes the direction and magnitude of the eccentricity of an orbit. It is named after Pierre-Simon Laplace, Carl Runge, and Wilhelm Lenz who all independently discovered its significance in the 18th and 19th centuries.

2. Why is the conservation of the Laplace-Runge-Lenz vector important?

The conservation of the Laplace-Runge-Lenz vector is important because it provides a mathematical proof for the stability of elliptical orbits in a central force field. It also allows scientists to accurately predict the motion of celestial bodies in our solar system and beyond.

3. What is the relationship between the Laplace-Runge-Lenz vector and the total energy of a system?

The Laplace-Runge-Lenz vector is directly related to the total energy of a system. It is a constant of motion and its magnitude is equal to the square root of the negative total energy. This relationship allows scientists to determine the energy of a system using the value of the Laplace-Runge-Lenz vector.

4. How is the Laplace-Runge-Lenz vector conserved?

The Laplace-Runge-Lenz vector is conserved because it obeys the conservation of angular momentum and the conservation of energy. This means that its magnitude and direction remain constant throughout the motion of a system, regardless of any external forces acting on it.

5. Can the conservation of the Laplace-Runge-Lenz vector be applied to quantum mechanics?

Yes, the conservation of the Laplace-Runge-Lenz vector has been extended to quantum mechanics and is known as the Runge-Lenz vector. It is a constant of motion in quantum systems and plays a significant role in the prediction and understanding of electron orbits in atoms and molecules.

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