Laplace-runge-lenz Vector

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	CATEGORIES ABOUT LAPLACE-RUNGE-LENZ VECTOR
	classical mechanics
	celestial mechanics
	rotational symmetry

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:$$ \mathbf{A}=\mathbf{p} imes \mathbf{L} - m k rac{\mathbf{r}}{r} where: $\backslash mathbf\{r\}\backslash !\backslash ,$ is the position vector of the mass $m\backslash !\backslash ,$, $\backslash mathbf\{L\}\backslash !\backslash ,$ is the Angular Momentum , $k\backslash !\backslash ,$ is a parameter that describes strength of the Potential . The Laplace-Runge-Lenz vector is constant for the motion of a particle acted on by a Central Force that varies as an inverse square (e.g., Gravity and Electrostatics ), i.e., for Potential s that vary as $-k/r\backslash !\backslash ,$. PROPERTIES By its definition, $\backslash mathbf\{A\}$ is perpendicular to $\backslash mathbf\{L\}$ (i.e., $$ \mathbf{A} \cdot \mathbf{L} = 0 ). (Recall that $\backslash mathbf\{r\}\; \backslash cdot\; \backslash mathbf\{L\}\; =\; 0$, because $\backslash mathbf\{L\}\; \backslash equiv\; \backslash mathbf\{r\}\; imes\; \backslash mathbf\{p\}$). The Laplace-Runge-Lenz vector can be used to derive the elliptical orbits of the Kepler problem :$$ \mathbf{A} \cdot \mathbf{r} \equiv Ar \cos heta = \mathbf{r} \cdot \left( \mathbf{p} imes \mathbf{L} ight) - mkr where $heta$ is the angle between the position and Laplace-Runge-Lenz vectors. Permuting the Triple Dot Product $\backslash mathbf\{r\}\; \backslash cdot\; \backslash left(\; \backslash mathbf\{p\}\; imes\; \backslash mathbf\{L\}\; ight)\; =\; \backslash mathbf\{L\}\; \backslash cdot\; \backslash left(\; \backslash mathbf\{r\}\; imes\; \backslash mathbf\{p\}\; ight)\; =\; L^\{2\}$ and rearranging yields the formula for an Ellipse :$$ rac{1}{r} = rac{mk}{L^{2}} \left( 1 + rac{A}{mk} \cos heta ight) The vector points toward the Pericenter , from the geometric center of the orbit to the attracting, central body. The magnitude for a Periodic Orbit with Eccentricity $e\backslash !\backslash ,$ is given by:

A simple prescription for quantizing a classical system is to set
the Commutation Relation s of the quantum mechanical operators
equal to the Poisson Bracket of the corresponding classical
variables, multiplied by $i\backslash hbar$. For brevity, we
define a reduced Laplace-Runge-Lenz vector

:$$