Potential and Kinetic Energy In engineering school you learn the basic swing problem. Essentially that there is a transfer of kinetic energy (as seen in the velocity at the bottom of as swing) to potential energy at the top of the swing's arc.
My question: What is the nature of KE and PE and how do they seamlessly "transfer" between the two? How is this explained by classical physics? 
 A: This is an attempt at an intuitive answer. Does it help?
A common form of energy is as a force applied to an object for a given
distance. This is typically what horses do when pulling a load, except that much is lost in friction, hence heat. In your case, if the swing is motionless in an up position and
you let go, it will go down because a force resulting from gravity is
applied to it and accelerate it to a given speed, for some height (the
distance). This force, acting on the way down (the distance) provide
the kinetic energy now stored in the speeding swing.  The swing, being
down, can no longer use that potential energy (gravity force pulling
down on a distance) to further accelerate (since the height/distance
has been used: the swing is down).
Actually, after passing the bottom part of its motion, the force
resulting from gravity will act again to slow it down as it is going
up, a negative acceleration, on the distance it take to go up. Hence
the swing loses speed and all its kinetic energy. But it is now up the
height it had before, and has regained the potential energy, i.e. the
possibility to accelerate by going down the same height difference as
before.
The real analysis is a bit more complicated because the motion of a
swing is circular. But the idea is there. The height is what matters here for potential energy.
A: Well I am not an expert on explaining stuff , here's my words for the question-
"What is the nature of KE and PE and how do they seamlessly "transfer" between the two?"
Lets take a ball that you are holding (I don't know why) in the middle of a road,its kinetic wrt you is zero.Again lets see that particle from from moving car reference frame,now its kinetics is wrt driver is "half mv square". 
What I am saying is that absolute kinetic energy is flawed same way as absolute potential. Whats matters is change.So change in kinetics energy is the quantitative amount of work that need to be done on the ball to make the change in velocity.
This work can be in any nature potential,frictional,internal etc. Potential work is in the category of conservative and others are non conservative (work is just a nature of force doing quantitative distance stuff.Note that every work indirectly comes from four forces)
So now potential: Potential (correctly Potential difference is change in arrangement of constituting entities in a Field Zone)  is put under conservative because it has a property that its sort of reversible with KE (you just need to change the arrangement) & the magic stuff that make this happen is a Field. So i believe the PE & KE talk to each other by directly or indirectly by fields.
So for applied Review purpose,you can yourself apply the above given information to your pendulum (Wait you were holding ball,sorry for that)
