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On the Principle of Least Action

Abstract. Much of the "mystery" of The Principal of Least Action is dispelled by the equation (<T>-<U>)<(<T>+DT>)-(<U>-DT>). 

In The Feynman Lectures on Physics, V1, Chapter 19, Richard Feynman states that, in any system conforming to F=ma, the average kinetic energy, <T>, minus the average potential energy,<U>, for the actual motion is always less than it would be for other, imagined motions. The value <T>-<U> is called the "Action." 

For reasons to be made clear, we shall amend that statement to read "In any system conforming to F=ma, the absolute value of the action for the actual motion is always less than it would be for other, imagined motions."

Let us consider the case of a mass m oscillating on an ideal spring of spring constant k. The amplitude of the oscillation is A, and the frequency is w=(k/m)1/2. The motion is along the x-axis with x=A sin(wt) and v=wA cos(wt). The kinetic energy, T, at any time is mv2/2, and the potential energy, U, is kx2/2. The total energy, E, equals T+U, and is constant throughout an oscillation period. 

The average kinetic energy, <T>, is the definite integral of (T)dt over the interval t=0 to t=2p/w, divided by t=2p/w. Similarly for the average potential energy, <U>.

The "action" of this system is <T> minus <U>. If we bias the kinetic energy by adding to it some function of x, and bias the potential energy by subtracting the same value from it, then we get a different, imagined motion. And the action for this system is always greater than it is for the real system. If on the other hand we bias the potential energy by adding some function of x to it, and adjust the kinetic energy by subtracting the same value, then we get a negative action. (That is why we should say that it is the absolute action (or the magnitude of the action), for the actual motion, that is always less than it would be for any imagined motion.)

Why should we add some bias to one variable, and subtract the same bias from the other? Because we wish to maintain E=T+U as a constant. And (T+DT)+(U-DT) still equals E.

There is nothing mysterious about the principle of least action. It is simply a consequence of the fact that, in the case of biased T for example, (<T>-<U>)<(<T>+DT>)-(<U>-DT>) for any <T>, <U> and DT. Similarly when we bias U.

In addition to classical mechanics, the Principle of Least Action plays important roles in Relativity, Quantum Theory, etc. Many interesting articles can be found on the Internet.