Ohne die Arbeit, part 3 (Long & wordy)

[Bob Sciamanda <sciamanda@worldnet.att.net>]

Some properties of U(r) and its relation to F(r)
		Submitted without proof (ask for details at your own peril.)
A conservative force was defined as a vector field F(r) whose line
integral through space is a function only of  the end points and
independent of the path chosen between those points.  Three adjunct
properties follow immediately: 
1) Changing the position of the reference point Rref simply adds a
constant to U(r) everywhere.  Since the MET involves only the delta U(r)
of two space points, the theorem is unaffected.  IOW U(r) is definable
and meaningful only to within an arbitrary additive constant; it is how
the value of U(r) varies from point to point which has physical import.
2) The line integral of a conservative force around any closed path
(identical start and finish points) is necessarily zero.
3) In a one dimensional space any f(x) is necessarily conservative.

At first blush it would seem that the "constant" frictional force f =
mu*N in a one dimensional problem would qualify under # 3) as a
conservative force.  Take the time to illustrate that this force does
not qualify because it is NOT describable as a f(x) since it changes
direction so as to always oppose the velocity vector and thus always
contribute a negative quantity to the LHS of the MET. Contrast that
behavior with the gravitational force mgh, which does not change
direction. Very instructive; do not omit.

After treating these simplest examples you should now consider the next
more involved F(x), a linear function. So integrate the Hooke's law
spring force to obtain U(x) = int(o=>x){(kx)dx} .  The integral to be
done here is again the area under a graph of x vs itself!  We can do
that!  U(x) = .5kx^2 . (It should be apparent that x is the elongation
of the spring beyond its relaxed length, x is negative for a
compression, and x=0 is the chosen reference point Rref for U=0.)

Staying in one dimension and treating a single mass under the influence
of  a conservative F(x), show that U(x) = -int{F(x)dx}  implies that
moving from x to x + dx will produce a change in U(x) given by dU(x) = -
F(x) dx  (extending the upper limit of the integral by dx simply adds
-Fdx to the area under the curve, and this is the change in U(x) .) 
This yields the important result: F(x) =  - dU(x)/dx .  This is the
inverse of the equation defining U(x) and allows one to recover F(x),
given U(x).  Note, importantly,  that the arbitrary additive constant in
U(x) does not enter into this calculation of F(x) from U(x).  The force
F(x) is equal to minus the space rate of change of the potential
function U(x), ie.; minus the slope of U(x). We should add this to our
above enumerated properties of U(r):
4) In a one dimensional space, F(x) = - dU(x)/dx 

Now you have the machinery to treat the Hooke's law oscillator using
graphs of U(x) and F(x) ; one the area under minus the other; the other
the negative of the slope of the first.  You can illustrate how the MET
(here a conservation statement) predicts the motion, the turning points,
etc.  You can add a frictional force which causes a "dissipation" of the
mechanical energy, and show how the turning points converge to the
stable equilibrium point at x = 0.  It is not forbidden at this point to
leap ahead to discussions of temperature rises, heat energy, and the
possibility, at least, of a wider energy conservation postulate.

> From the Hooke's law oscillator, generalize to other one dimensional
potential functions, illustrate stable and unstable equilibrium points,
escape velocities, etc. - there is no end, even in just one dimension!

The use of potential energy functions will usually be restricted to
problems involving a single particle or a single rigid body,  because
the conservative force must be a function of the CM position only. 
However, one can sometimes rework the model description so that the
mathematics is the same as if the above were true, even though it was
not true in the original model. As a very useful example consider the
dumbbell system of two masses interacting through a central force
F(|R|); R is the vector locating mass 1 from mass 2 and F(|R|) depends
only on its magnitude.  This could be the model for a variety of
physical systems, from a hydrogen molecule to a binary star system.  We
are interested in the time behavior of the vector R, ie.; the motion of
M1 relative to M2, assuming the dumbbell system is isolated from other
forces.

		  R2				R1
 
M2<-----------------------CM----------------------------------------------------->M1
	 
------------------------------------------------------------------------------->
					R
The difficulty is that if we define M1 as our system, the force on it
depends not only on its location, but also on the location of M2, which
will not stand still for us. However:
> From the definition of the CM,   M1*R1 + M2*R2 = 0			(Eq #1)


By construction,  R = R1 - R2 = R1*(1 + M1/M2)  , after using Eq
#1.         	(Eq #2)

Since the CM is an inertial origin,  M1*R1'' = F(|R|) * R / |R|	; ('' =
2nd time derivative)  (Eq#3)

Using Eq#2,  m*R'' = F(|R|) * R / |R| , where m = (M1*M2)/(M1+M2), the
"reduced" mass.  (Eq #4)

Eq (4) describes the behavior of the vector R, which locates M1 from
(the moving) M2, and says that its behavior is the same as that of a
particle of mass m under the influence of the central force F(|R|) of a
FIXED source.  Obviously, the MET can use a potential energy function to
describe the behavior of R in Eq #4.      More to come . . .

-- 


Bob Sciamanda				sciamanda@edinboro.edu
Dept of Physics				sciamanda@worldnet.att.net
Edinboro Univ of PA			http://www.edinboro.edu/~sciamanda/home.html
Edinboro, PA				(814)838-7185

In the fall of 1972 President Nixon announced that the rate of increase
of inflation was decreasing. This was the first time a sitting president
used the third derivative to advance his case for reelection.
Hugo Rossi, Mathematics Is an Edifice, Not a Toolbox, Notices of the
AMS, v. 43, no. 10, October 1996.