The Mechanical Energy Theorem (MET) states that, for any system of
particles and objects, the line integral of the net external force over
the trajectory of the system CM is numerically equal to the change in
the CM kinetic energy. It should seem remarkable to the sensitive
physicist that the RHS of this equality depends only on the endpoints of
the motion; ie., it does not matter how the CM velocity V(t) varies in
time, only the end values of V(t) are used.
(Aside): When, as a teacher, you derive the MET you should pause and
marvel at how this happens when you integrate the RHS of Fnet (dot) dRcm
=M(dV/dt) (dot) dRcm. Here V, dV/dt, and Rcm are all changing in time
and it looks very much like one needs to know the details of the motion
before the RHS can be integrated. However, the integral of the RHS is
easily turned into the sum of three terms of the form M*int{Vi * dVi},
where wazzu Vi is a cartesian component of the CM velocity.
Remarkably, each of these definite integrals is simply the area under a
graph of Vi vs. itself! Thus each term is just M*(.5Vi^2)2 minus
M*(.5Vi ^2)1; ie., M times the difference in the area of two triangles,
no matter how Vi varies in time! One does not need a calculus course to
do this, only an appreciation of the definite integral int{y(x)*dx} as
the area under a plot of y(x) vs x. (Ask for any required details.)
Wouldn't it be cool if the RHS of the MET, int{Fnet (dot) dRcm}, could
also be evaluated by looking at only the end points of the trajectory!
Indeed, this can be done for a certain, special kind of force. Suppose
one of the external forces depends only on the position of the system
CM, so that we can write Fnet = F(Rcm) + F_other, and suppose further
that the line integral of F(Rcm ) over any trajectory depends only on
the coordinates of the end points of that trajectory and not at all on
the particular path chosen between those end points. (For reasons which
will become apparent, we shall call such a special force a "conservative
force".) One can then choose an arbitrary reference point (Rref) and
define a scalar field U(R) which assigns to each point in space a
number. That number will be minus the line integral of this
conservative force from the reference point Rref to the general point R
(Note that such a U(R) will be uniquely definable only for a
conservative force.) It is then easily shown that the contribution of
F(Rcm) to the line integral of Fnet (dot) dRcm is numerically equal to
U(R1) minus U(R2). The MET then can be written as:
line int{F_other (dot) dRcm} +U(R1) - U(R2) = CMKE_2 - CMKE_1 . This
is conventionally written:
line int{F_other (dot) dRcm} = CMKE_2 + U(R2) - CMKE_1 - U(R1) , or
line int{F_other (dot) dRcm} = ME_2 - ME_1, where ME_i = CMKE_i +
U(Ri)
Since each term has units of energy, we may call U(R) the potential
energy function" (or simply the potential energy), and ME the "total
mechanical energy". In the general case, one may include in ME a
potential energy term Ui(R) for each conservative force in Fnet. In the
fortuitous case where all of the forces are conservative, then F_other =
zero, and we have:
0 = ME_2 - ME_1, or ME_1 = ME_2, or ME = constant, or CMKE + U1(R)
+U2(R)+ ... =constant.
So that if only conservative forces contribute to the line integral of
Fnet (dot) dRcm, we can assert a conservation statement: The grand sum
of the center of mass kinetic energy and all of the potential energy
functions is a constant of the motion.
For the general case, when F_other is not zero, we may assert:
So that if there are other forces acting, not represented by a U(R)
function, their line integral will be numerically equal to the change
in the system's total mechanical energy. This last equality is what I
have always taught as the *bleep* - energy theorem in its most general
(translational) form . I propose it be re-named the Mechanical Energy
Theorem (MET) and that the LHS be referred to simply as the CM line
integral of F_other.
It should be apparent that if there are no conservative forces, then
F_other = Fnet, ME = CMKE, and we are back to the original statement
of the MET: Int{Fnet (dot) dRcm} = delta (CMKE). This will also be
our statement of the MET if there are conservative forces, but we do not
choose to replace their line integrals with the potential functions.
Note that a potential energy function U(R) is here simply another way of
calculating the line integral of a conservative force, so that no more
should be read into the Ui(R) than can be read into the line integrals
of the forces, simply on the basis of their appearance in the MET. Any
further interpretation (eg., as particular forms of some universally
conserved quantity) will have to come from postulates of a wider scope
and completely different kind, eg., about energy, its various forms and
its conservation, such as are made in the first law of thermodynamics
(FLT).
But even without additional assumptions there lies a rich mathematical
treasure in the further study of the scalar potential field U(r) and its
relation to the vector force field F(r).
more to come . . .
-Bob
--
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
The successes of the differential equation paradigm were impressive and
extensive. Many problems, including basic and important ones, led to
equations that could be solved. A process of self-selection set in,
whereby equations that could not be solved were automatically of less
interest than those that could.
Stewart, Ian, Does God Play Dice? The Mathematics of Chaos. Blackwell,
Cambridge, MA, 1989, p. 39.