## Runge-Lenz vector conservation

Posted by peeterjoot on February 13, 2012

# Motivation.

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.

# Runge-Lenz vector

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

For

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

With

our vector must be some constant vector. Let’s write this

so that

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

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

or

Our dynamics are now fully specified, even if this not completely explicit

What we can do is rearrange and separate variables

to find

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

Or

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).

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