# 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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### Energy eigenvalue of hydrogen-like atoms using Laplace-Runge-Lenz vector

I have a basic question about a few calculations involving the quantum mechanical Laplace-Runge-Lenz vector. In classical mechanics there is the Laplace-Runge-Lenz vector, which for a hydrogen-like ...
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### What exactly are the 12 conserved quantities in the Two-Body Problem?

The Two-Body problem consists of 6 2nd-order differential equations \ddot{\mathbf{r}}_1 = \frac{1}{m_1}\ \mathbf{F_g} \\ \ddot{\mathbf{r}}_2 = -\ \frac{1}{m_2}\ \mathbf{F_g} \end{...
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### Runge-Lenz vector when the earth is neither at the aphelion nor at the perihelion

Assuming the earth is either at the perihelion or at the aphelion, it is easy to see the Runge-Lenz (RL) vector is directed along the line joining the perihelion and aphelion. Since the RL vector is a ...
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### Conserved Quantities in Kepler Problem?

In our classical mechanics class, professor said that Kepler's problem is a kind of Integrable System such that the number of conserved quantities would be equal to the number of degrees of freedom. ...
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### Why Kepler problem is equivalent to a free particle on 4 dimensional sphere?

In trying to understand Laplace-Runge-Lenz vector, I read in Wikipedia that the Kepler problem is mathematically equivalent to a particle moving freely on the surface of a four dimensional hypersphere....
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### Quantum Analog to Kepler's First Law

According to Kepler's First Law, the orbit of a planet is an ellipse round the sun with the sun at one focus. There's an inherent asymmetry in this. Instead of the sun being in the dead center, its ...
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### Component-free computation of Poisson bracket of Laplace-Runge-Lenz vector

How can the Poisson bracket $\{A,H\}$ be computed directly without components, where $H$ is the Hamiltonian for the inverse square force, $$H=\frac{p^2}{2m} - \frac{k}{|r|}\ ,$$ and $A$ is the ...
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### Is the Laplace-Runge-Lenz vector applicable for test particle motion around black holes?

In classical mechanics , the Laplace-Runge-Lenz (LRL) vector is a characteristic feature of the Kepler problem. This enables a very simple discussion of the properties of the orbit for the problem. It ...
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### How to relate Laplace-Runge-Lenz vector to eccentricity?

So the eccentricity can be written in this form $$\mathbf{e}=\frac{\mathbf{A}}{mk} =\frac{1}{mk}(\mathbf{p}\times\mathbf{L})-\hat{\mathbf{r}}$$ but I cannot find a proof or figure it out on my own.
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### Question about Sakurai's $SO(4)$ symmetry section

In Sakurai's Quantum mechanics book, he says the hydrogen atom has $SO(4)$ symmetry by explicitly exhibiting operators $I_i,K_i$ that satisfy the commutation relation of the Lie algebra $so(4)$. ...
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### Physical implication of conservation of Laplace-Runge-Lenz vector

I'm having trouble wrapping my mind around the Laplace-Runge-Lenz vector. Conservation of momentum can be visualized as an object moving in a straight line with constant speed. One can even visualize ...
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### Showing The Laplace–Runge–Lenz vector (per unit mass) is constant

Given an inverse square law $\ddot{\vec{r}}=-\frac{\mu}{r^2}\hat{r}$, I define the Angular momentum per unit mass as $\vec{H}=\vec{r}\times\dot{\vec{r}}$. Showing it's constant is strightfoward. Then ...
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It can be proven that the commutation relations $$\Big[ N,H\Big]=0$$ $$\Big[ L,H\Big]=0$$ $$\Big[ N,L^2\Big] \neq 0$$ Where $N$ is the Runge-Lenz vector. I'm wondering if it is possible to write ...