Laplace-Runge-Lenz vector and its generating transformation

  • Thread starter Sonty
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I have this sort of research project about symmetries under the central potential and I'm stuck on this Runge-Lenz vector. As it is a conserved quantity I was expecting it to come out of Noether's theorem. I can't figure out how. So I go on the net to find out and get 2 answers: infinitesimal Lorentz trasnformation without rotation followed by a time translation and a more explicit article on a canonical transformation. Out of the latter I find that besides the usual space-time transformations come the so called kinematical symmetries, while out of symmetries in the phase space comes another kind of conserved quantities called dynamical. It shows rather clearly that the angle the LRL vector makes with the x axis is the canonical conjugate of L and that's the pair of variables I should canonical transform to.
What other dynamical symmetries are there in the world that I heven't heared of and how should the conserved quantities look in general terms? Is there an analogue to Noether's theorem that would tell me exactly how the conserved quantity should look like? How did Laplace, Runge and Lenz get the expression for the vector?
 
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that's the first result in google's search and the second in teoma's. It doesn't help. It's like trees are green, the sky is blue, the water is wet, stuff like that. It doesn't say why or how.
 
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Originally posted by Sonty
that's the first result in google's search and the second in teoma's. It doesn't help. It's like trees are green, the sky is blue, the water is wet, stuff like that. It doesn't say why or how.
Uh, have you actually read the paper cited by that message, the one in Phys Rev?

The abstract:

Noether and Lie symmetry analyses based on point transformations that depend on time and spatial coordinates will be reviewed for a general class of time-dependent Hamiltonian systems. The resulting symmetries are expressed in the form of generators whose time-dependent coefficients follow as solutions of sets of ordinary differential ("auxiliary") equations. The interrelation between the Noether and Lie sets of auxiliary equations will be elucidated. The auxiliary equations of the Noether approach will be shown to admit invariants for a much broader class of potentials, compared to earlier studies. As an example, we work out the Noether and Lie symmetries for the time-dependent Kepler system. The Runge-Lenz vector of the time-independent Kepler system will be shown to emerge as a Noether invariant if we adequately interpret the pertaining auxiliary equation. Furthermore, additional nonlocal invariants and symmetries of the Kepler system will be isolated by identifying further solutions of the auxiliary equations that depend on the explicitly known solution path of the equations of motion. Showing that the invariants remain unchanged under the action of different symmetry operators, we demonstrate that a unique correlation between a symmetry transformation and an invariant does not exist.
 
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I had it but didn't fully read it. I had just skimmed through it searching for keywords. Didn't find them until I actually read a big part of it. In my reaction I confused it for another one. The thing is that seems to me to elaborate. I don't think that's what I'm supposed to get. I couldn't follow his calculations fully, maybe because I'm tired and I've been fighting this nasty flu for a couple of days. There's got to be an easier way to do it.
 

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