= Laplace-Runge-Lenz vector
{c}
{title2=$\mathbf A$}
{wiki=Laplace–Runge–Lenz_vector}
= Runge-Lenz vector
{c}
{synonym}
For the signed <inverse-square force> $\boldsymbol F=-\kappa\boldsymbol r/r^3$ on a particle of <mass> $m$, with Hamiltonian $H=p^2/(2m)-\kappa/r$, the <Laplace-Runge-Lenz vector> $\boldsymbol A=\boldsymbol p\times\boldsymbol L-m\kappa\widehat{\boldsymbol r}$ is conserved. Attraction has $\kappa>0$ and repulsion has $\kappa<0$. It obeys $\boldsymbol A\cdot\boldsymbol L=0$ and $A^2=m^2\kappa^2+2mHL^2$. For unit <mass> in a <Kepler orbit>, $\kappa=GM$.
Back to article page
