Editing Laplace–Runge–Lenz vector (section)

=== Unscaled functions ===
The algebraic structure of the problem is, as explained in later sections, {{math|1=SO(4)/'''Z'''<sub>2</sub> ~ SO(3) × SO(3)}}.<ref name="bargmann_1936" />
The three components ''L<sub>i</sub>'' of the angular momentum vector {{math|'''L'''}} have the Poisson brackets<ref name="goldstein_1980" />
<math display="block">
\{ L_i, L_j\} = \sum_{s=1}^3 \varepsilon_{ijs} L_s,
</math>
where {{mvar|i}}=1,2,3 and {{math|''ε<sub>ijs</sub>''}} is the fully [[antisymmetric tensor]], i.e., the [[Levi-Civita symbol]]; the summation index {{mvar|s}} is used here to avoid confusion with the force parameter {{mvar|k}} defined [[#Mathematical definition|above]]. Then since the LRL vector {{math|'''A'''}} transforms like a vector, we have the following Poisson bracket relations between {{math|'''A'''}} and {{math|'''L'''}}:<ref>{{harvnb|Hall|2013}} Proposition 17.25.</ref>
<math display="block">\{A_i,L_j\}=\sum_{s=1}^3\varepsilon_{ijs}A_s.</math> 
Finally, the Poisson bracket relations between the different components of {{math|'''A'''}} are as follows:<ref>{{harvnb|Hall|2013}} Proposition 18.7; note that Hall uses a different normalization of the LRL vector.</ref>
<math display="block">\{A_i,A_j\}=-2mH\sum_{s=1}^3\varepsilon_{ijs}L_s,</math>
where <math>H</math> is the Hamiltonian. Note that the span of the components of {{math|'''A'''}} and the components of {{math|'''L'''}} is not closed under Poisson brackets, because of the factor of <math>H</math> on the right-hand side of this last relation.

Finally, since both {{math|'''L'''}} and {{math|'''A'''}} are constants of motion, we have
<math display="block">\{A_i, H\} = \{L_i, H\} = 0.</math>

The Poisson brackets will be extended to quantum mechanical [[canonical commutation relation|commutation relations]] in the [[#Quantum mechanics of the hydrogen atom|next section]] and to [[Lie algebra|Lie bracket]]s in a [[#Conservation and symmetry|following section]].