From Jeff Findley:
In article ,
says...

What I don't understand is that the point where the satellite starts to
go faster than needed for that altitude happens before perigee. How
come it continues to drop even if it is going faster than needed to
remain in that orbital altitude?


Because it is not going faster IN THE RIGHT DIRECTION. So it
continues to drop and gain speed until its velocity IN THE RIGHT
DIRECTION is too high, at which point it starts going back up and
slowing down.


So what is magical about perigee that causes the satellite who is
already going way faster than necessary to finally stop losing
altitude/accelerating and starts to behave normally for a satellite that
is going faster than needed at that altitude? (gain altitude, lose speed)


Resolve the velocity into two components, one tangential to a circular
orbit and one normal to that. When your tangential velocity exceeds
orbital speed you start going back up.


This. You have to use vector math to analyze orbital mechanics, not
scalar math. For a two body problem, the motion is at least planar,
which reduces the complexity to a 2D vector math problem.

As was posted yesterday to this thread, it was offered that 2-body orbit dynamics can be approximated by using a 1D spring-mass model. So that says that vector math is *not necessary* in order to grasp the basics of what is happening in circular and elliptical orbits.

And for the circular orbit case, the motion further reduces to Zero-dimensions (0D). The satellite (or moon, planet, star, what have you) just sits there absolutely still (in a rotating coordinate frame of reference).

~ CT