Editing Laplace–Runge–Lenz vector (section)

=== Scaled functions ===
As noted [[#Alternative scalings, symbols and formulations|below]], a scaled Laplace–Runge–Lenz vector {{math|'''D'''}} may be defined with the same units as angular momentum by dividing {{math|'''A'''}} by <math display="inline">p_0 = \sqrt{2m|H|}</math>. Since {{math|'''D'''}} still transforms like a vector, the Poisson brackets of {{math|'''D'''}} with the angular momentum vector {{math|'''L'''}} can then be written in a similar form<ref name="bargmann_1936" /><ref name="bohm_1993" />
<math display="block">
\{ D_i, L_j\} = \sum_{s=1}^3 \varepsilon_{ijs} D_s.
</math>

The Poisson brackets of {{math|'''D'''}} with ''itself'' depend on the [[sign (mathematics)|sign]] of {{mvar|H}}, i.e., on whether the energy is [[negative number|negative]] (producing closed, elliptical orbits under an inverse-square central force) or [[positive number|positive]] (producing open, hyperbolic orbits under an inverse-square central force). For ''negative'' energies—i.e., for bound systems—the Poisson brackets are<ref name="Hall 2013">{{harvnb|Hall|2013}} Theorem 18.9.</ref>
<math display="block">
\{ D_i, D_j\} = \sum_{s=1}^3 \varepsilon_{ijs} L_s.
</math>
We may now appreciate the motivation for the chosen scaling of {{math|'''D'''}}: With this scaling, the Hamiltonian no longer appears on the right-hand side of the preceding relation. Thus, the span of the three components of {{math|'''L'''}} and the three components of {{math|'''D'''}} forms a six-dimensional Lie algebra under the Poisson bracket. This Lie algebra is isomorphic to {{math|so(4)}}, the Lie algebra of the 4-dimensional rotation group {{math|SO(4)}}.<ref name="ReferenceA">{{harvnb|Hall|2013|at=Section 18.4.4.}}</ref>

By contrast, for ''positive'' energy, the Poisson brackets have the opposite sign,
<math display="block">
\{ D_i, D_j\} = -\sum_{s=1}^3 \varepsilon_{ijs} L_s.
</math>
In this case, the Lie algebra is isomorphic to {{math|so(3,1)}}.

The distinction between positive and negative energies arises because the desired scaling—the one that eliminates the Hamiltonian from the right-hand side of the Poisson bracket relations between the components of the scaled LRL vector—involves the ''square root'' of the Hamiltonian. To obtain real-valued functions, we must then take the absolute value of the Hamiltonian, which distinguishes between positive values (where <math>|H| = H</math>) and negative values (where <math>|H| = -H</math>).