Notes from Prof. Poppitz’s phy354 classical mechanics lecture on the Runge-Lenz vector, a less well known conserved quantity for the 3D potentials that can be used to solve the Kepler problem.
Motivation: The Kepler problem.
We can plug away at the Lagrangian in cylindrical coordinates and find eventually
but this can be messy to solve, where we get elliptic integrals or worse, depending on the potential.
For the special case of the 3D problem where the potential has a form, this is what Prof. Poppitz called “super-integrable”. With conserved quantities to be found, we’ve got one more. Here the form of that last conserved quantity is given, called the Runge-Lenz vector, and we verify that it is conserved.
Given a potential
and a Lagrangian
and writing the angular momentum as
the Runge-Lenz vector
is a conserved quantity.
Verify the conservation assumption.
Let’s show that the conservation assumption is correct
Here, we note that angular momentum conservation is really , so we are left with only the acceleration term, which we can rewrite in terms of the Euler-Lagrange equation
We can compute the double cross product
Plugging this we have
Now let’s look at the other term. We’ll need the derivative of
Putting all the bits together we’ve now verified the conservation statement
our vector must be some constant vector. Let’s write this
Dotting 3.12 with we find
With lying in the plane of the trajectory (perpendicular to ), we must also have lying in the plane of the trajectory.
Now we can dot 3.12 with to find
This is a kind of curious implicit relationship, since is also a function of . Recall that the kinetic portion of our Lagrangian was
so that our angular momentum was
with no dependence in the Lagrangian we have
Our dynamics are now fully specified, even if this not completely explicit
What we can do is rearrange and separate variables
Now, at least is specified implicitly.
We can also use the first of these to determine the magnitude of the radial velocity
with this, we can also find the energy
Is this what was used in class to state the relation
It’s not obvious exactly how that is obtained, but we can go back to 3.18 to eliminate the term
Presumably this simplifies to the desired result (or there’s other errors made in that prevent that).