Kepler’s second law for such motion comes from conservation of angular
momentum, which corresponds to the Poisson bracket relation
{lj,h}= 0
Here we’ll take the Coulomb version of the Hamiltonian that we need for the
hydrogen atom problem
h=
1
2 m
|p|^2 −
e^2
r
The relation{lj,h}= 0 can be read in two ways:
- The Hamiltonianhis invariant under the action of the group (SO(3))
whose infinitesimal generators arelj. - The components of the angular momentum (lj) are invariant under the
action of the group (Rof time translations) whose infinitesimal generator
ish, so the angular momentum is a conserved quantity.
For this special choice of Hamiltonian, there is a different sort of conserved
quantity. This quantity is, like the angular momentum, a vector, often called
the Lenz (or sometimes Runge-Lenz, or even Laplace-Runge-Lenz) vector:
Definition(Lenz vector).The Lenz vector is the vector-valued function on the
phase spaceR^6 given by
w=
1
m
(l×p) +e^2
q
|q|
Simple manipulations of the cross-product show that one has
l·w= 0
We won’t here explicitly calculate the various Poisson brackets involving the
componentswjofw, since this is a long and unilluminating calculation, but
will just quote the results, which are
- {wj,h}= 0
This says that, like the angular momentum, the vector with components
wjis a conserved quantity under time evolution of the system, and its
components generate symmetries of the classical system.
{lj,wk}=jklwl
These relations say that the generators of theSO(3) symmetry act onwj
in the way one would expect for the componentswjof a vector inR^3.