# Questions tagged [laplace-runge-lenz-vector]

The Laplace–Runge–Lenz vector describes the shape and orientation of the orbit of one astronomical body around another. In general, the LRL vector is conserved (it's a constant of the motion) in all problems in which two bodies interact by a central force that varies as the inverse square of the distance between them (Kepler problem). Its conservation is significant in the quantization of the Hydrogen atom.

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### Why Laplace-Runge-Lenz vector in a circular motion is $0$?

This vector is the sum of two vectors, and I understand why their direction is opposite, but I don't understand why their magnitude is the same. I know that the direction of this vector is always the ...
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### Hydrogen atom as a free particle moving on a three dimensional sphere?

I have heard on various occasions that the Hamiltonian/Lagrangian of the Hydrogen atom or that of a particle moving in $1/r$-potential can be transformed into that of a free particle moving on the ...
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### How to see the ${\rm SO}(4)$ symmetry of the classical Kepler problem?

It is well-known that the hidden symmetry of the $1/r$-problem $$H=\frac{{\bf p}^2}{2m}-\frac{k}{r}\tag{1}$$ is ${\rm SO(4)}$ in the sense that the components of angular momentum $L_i$ and those of ...
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This is part of an admission exam problem, found at http://www.sissa.it/mp/admission/tests//2008_common.pdf Consider the Hamiltonian of Kepler problem $$H(\boldsymbol{r},\boldsymbol{p})= \frac{|\... • 21 2 votes 0 answers 1k views ### Square of Laplace–Runge–Lenz vector in Hydrogen atom [closed] I have a problem. I've tried this question, but I don't get the correct expression. Can someone give me some ideas? Thanks! Consider the Hydrogen Atom Hamiltonian:$$ H = (\mathbf p^2/2 \mu)-(e^2/...
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I have been trying for hours and cannot figure it out. I am not asking anyone to do it for me, but to understand how to proceed. We have the relations $$[L_i,p_j] ~=~ i\hbar\; \epsilon_{ijk}p_k,$$ ...
Noether's theorem relates symmetries to conserved quantities. For a central potential $V \propto \frac{1}{r}$, the Laplace-Runge-Lenz vector is conserved. What is the symmetry associated with the ...