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**=m**a**,
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**=m**a**,
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 mv^{2}/2, and the potential energy, U, is kx^{2}/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.