= Kepler dynamical symmetry algebra
{c}
{title2=$\mathfrak{so}(4),\quad\mathfrak e(3),\quad\mathfrak{so}(3,1)$}
For $H=p^2/2-\kappa/r$, the <angular momentum> and <Laplace-Runge-Lenz vector> obey $\{K_i,K_j\}=-2H\epsilon_{ijk}L_k$. On negative or positive energy regions use $K/\sqrt{|2H|}$ to obtain the compact or Lorentz <Lie algebra>, respectively. At zero energy the algebra on the characteristic orbit quotient is the Euclidean motion algebra. Global <group actions> require complete generators or collision regularization; simply replacing $H$ by a constant inside a Hamiltonian-vector-field derivative is not valid on the unreduced energy hypersurface.
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