Symmetry behind Laplace-Runge-Lenz vector conservation?

In summary, for gravitation and any inverse-square forces, the Laplace-Runge-Lenz vector is conserved. This is a consequence of the 1/r² dependence of the force and the spherical symmetry, which leads to the conservation of angular momentum. The LRL vector is defined as A=p×L - mαe_r, where p and L are the linear and angular momentum, respectively, and mαe_r is a constant. As both p×L and mαe_r are conserved quantities, the LRL vector is also conserved. While there may not be a specific symmetry associated with the LRL vector, it is formed from the product of two conserved quantities and can be seen as a combination
  • #1
lalbatros
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2
For gravitation and any inverse-square forces, the Laplace-Runge-Lenz vector is conserved. (see http://en.wikipedia.org/wiki/Laplace-Runge-Lenz_vector" [Broken])
Any conserved quantity is associated with a symmetry of the Hamiltonian with respect to some coordinate, according to the Noether theorem.

I would like to know what this symmetry coordinate(s) is (are) when the conservation of the LRL vector is involved.

Thanks
 
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  • #2
lalbatros said:
For gravitation and any inverse-square forces, the Laplace-Runge-Lenz vector is conserved. (see http://en.wikipedia.org/wiki/Laplace-Runge-Lenz_vector" [Broken])
Any conserved quantity is associated with a symmetry of the Hamiltonian with respect to some coordinate, according to the Noether theorem.

I would like to know what this symmetry coordinate(s) is (are) when the conservation of the LRL vector is involved.

Thanks

i don't know if i got right your question, but i think that the answer is spherical coordinate, since H=P^2+V(r), and you have anly radial dependence on the V.
 
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  • #3
Marco_84,

The spherical symmetry is not enough to imply the conservation of the LRL vector.
The conservation of the LRL is a consequence of the 1/r² dependence of the force.
But what is then the symmetric coordinate(s) behind this conservation law?

This might also be partly related to the fact that the trajectories are closed which only happen for 1/r² and r² potentials. Similarly, other dependences (1/r4, ...) lead to non-closed trajectories.
 
  • #4
lalbatros said:
Marco_84,

The spherical symmetry is not enough to imply the conservation of the LRL vector.
The conservation of the LRL is a consequence of the 1/r² dependence of the force.
But what is then the symmetric coordinate(s) behind this conservation law?

This might also be partly related to the fact that the trajectories are closed which only happen for 1/r² and r² potentials. Similarly, other dependences (1/r4, ...) lead to non-closed trajectories.
yes i think you are right, the spherical symmetry gives you angular momentum conservation. But if you're system is kepler's one you get also that Lenz vector is conserved, in fact in the proof you use only this 2 facts...
and the coordinate you use is the only radial one since the forces are central.
since we live in 3-dim world the dependence on V is r^-2, but we colud have log(r) depndence in flat world... i think that's why we find This vector.
a good exercice is to calculate what kind of vector we find in other dimension...

bye marco
 
  • #6
siddharth,

You are right, it looks like an answer, but I have two problems with it:

1) I don't understand it
2) I want to relate the conservation of the LRL vector to a symmetry and the Noether theorem

I don't see how the scaling symmetry can fit in the Noether theorem and answer my question, since the hamiltonian is changed by this scaling. But since the scaling of the Hamiltonian is known and simple, maybe the answer might reduce to a tiny extension of the Noether theorem or to small re-definition of the Kepler Hamiltonian so as to restore a perfecty symmetry and make the Noether theorem directly applicable (without changing any physics).

There is probably a simple answer.

(note: I should check how the scaling affects the motion)
 
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  • #7
The answer is much simplier than you can think. The Lenz-Vector is defined as follows:

A=p[tex]\timesL[/tex] - m[tex]\alpha[/tex]e[tex]_{r}[/tex]

m[tex]\alpha[/tex]e_{r} is a constant. Therefore

[tex]\frac{d}{dt}[/tex] m[tex]\alpha[/tex]e_{r}=0

As p[tex]\timesL[/tex] is a constant too,

[tex]\frac{d}{dt}[/tex]A=p[tex]\timesL[/tex] - m[tex]\alpha[/tex]e[tex]_{r}[/tex]=0

shows that the Lenz-Vector is preserved.

There is no symmetry behind it itself, since it is formed from the product of angular and linear momentum and their preservation is derived from their symmetries.

(sorry for the formatation)
 
  • #8
Well observed D'Alembert !
It is clear that any combination of several constants of motion is also a constant of motion.
Obviously, one could generate many related constant of motion, and each of them do not need to be associated with a symmetry.
So, the Noether theorem goes from symmetry to a constant, and not the reverse.

Thanks for the observation!
 

1. What is the significance of the Laplace-Runge-Lenz vector in symmetry conservation?

The Laplace-Runge-Lenz vector is a mathematical quantity that characterizes the symmetry properties of a physical system. It is a conserved quantity, meaning that it remains constant over time, and is closely related to the conservation of energy and angular momentum in a system. Its presence in a system indicates that the system has a specific type of symmetry known as central symmetry.

2. How does the Laplace-Runge-Lenz vector relate to the laws of motion?

The Laplace-Runge-Lenz vector is closely related to the laws of motion, specifically the conservation of angular momentum. It arises from the fact that the laws of motion are invariant under certain transformations, including translations and rotations. The presence of the Laplace-Runge-Lenz vector in a system indicates that the system has a specific type of symmetry, known as central symmetry, which is related to the laws of motion.

3. Can the Laplace-Runge-Lenz vector be applied to all physical systems?

No, the Laplace-Runge-Lenz vector is only applicable to systems that exhibit central symmetry. This means that the forces acting on the system must be directed towards a single point, known as the center of symmetry. Examples of systems that exhibit central symmetry include planetary orbits and the motion of charged particles in an electric field.

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

The Laplace-Runge-Lenz vector is calculated using the positions and momenta of the particles in a system. It is given by the cross product of the position vector and the momentum vector, and is then multiplied by the mass of the particle. In other words, the Laplace-Runge-Lenz vector is equal to the angular momentum of the particle multiplied by the mass and divided by the reduced mass of the system.

5. What are the practical applications of the Laplace-Runge-Lenz vector?

The Laplace-Runge-Lenz vector has many practical applications in physics, particularly in the study of celestial mechanics and quantum mechanics. It has been used to accurately predict the orbits of planets and other celestial bodies, and has also been applied to the study of atomic and subatomic particles. Its conservation also has implications for the stability and symmetry of physical systems, making it an important concept in many areas of physics.

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