Constant energy in an elliptical orbit

[Sleepycoaster]

Homework Statement

There's no specific question, but mostly a theory I wanted clarified. According to my textbook, the measurement of the total mechanical energy E of a mass orbiting a much larger mass in an ellipse is:

E = radial (change in radius) kinetic energy + rotational kinetic energy + potential energy

Or in other words,

E = (1/2)μ(dr/dt)² + (1/2)(angular momentum)^2 / μr² - GMm/r

But consider this: At both the apogee and perigee of the orbit the total energy should be constant and radial kinetic energy should be zero. Yet at these points the r is different and everything else is the same. How can energy be constant?

Homework Equations

r is the radius between the focus of the ellipse where the larger mass is and the smaller mass
μ is the reduced mass 1/(1/m + 1/M)

The Attempt at a Solution

None so far. It seems like a rather fundamental issue with a hopefully fundamental fix.

 

[TSny]

Hello, sleepycoaster.

E = (1/2)μ(dr/dt)² + (1/2)(angular momentum)^2 / μr² - GMm/r

But consider this: At both the apogee and perigee of the orbit the total energy should be constant and radial kinetic energy should be zero.

Yes.

Yet at these points the r is different and everything else is the same. How can energy be constant?

Note that r occurs in the denominator of two terms and the two terms have opposite signs.