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results on the four hypothetical orbits, plus his result of
the Kepler problem, because my force balance method is
the inverse of his method. He posed his orbital problems
in the following way. Given the shape of the orbit and
the location of the force center, find the functional form
of the central attractive force that will keep a body in
this orbit. My method is [5]: given the functional form of
the central force and the force balance, find the shape of
the orbit and its relation to the force center.
The Kepler problem is: for an elliptical orbit with the
force center at one focus of the ellipse, find the form of
the central attractive force. Newton obtained the inverse
square law for the force in this case, i.e. the force varies
as the inverse square of the distance between the focus
and the body. My force balance method puts the inverse
square law in and comes out with an elliptical orbit with
the force center at one focus.
Four hypothetical orbital problems worked out in the
Principia have not led to any practical applications so far,
with one recent exception: the elliptical orbit with the
force center at the center of the ellipse. Here Newton
proved that the attractive force varies directly as the dis-
tance, like a linear spring, which is a special case of
Hooke’s Law. To the best of my knowledge Newton
never used the term “spring” in connection with this hy-
pothetical problem in the Principia, but it is known that
Hooke and Newton were not the closest of colleagues.
Recently a practical application of this orbital problem to
surface gravity waves has been found, as discussed in
Section 6.
By putting Newton’s results for the central forces, one
at a time, into my force balance equation in polar coor-
dinates, that includes the centrifugal force, I can easily
get the differential equations to be solved for the shapes
of the orbits. In all cases the left side of the equation is
the linear harmonic oscillator equation in polar coordi-
nates where the variable of oscillation is the inverse of
the radius (which may seem a bit unusual). Depending
on the form of the force, the right side of the force bal-
ance equation can be linear or more generally nonlinear.
Then if the corresponding shapes of the orbits that New-
ton began with are inserted into the differential equations,
these equations can be solved and complete consistency
is found between the two independent methods for all
four hypothetical orbital problems in the Principia as
well as for the Kepler problem. One striking offshoot is
that analytic solutions in closed form, in terms of ele-
mentary functions, exist for two different differential
equations that are both fully nonlinear, which one would
never expect to find (one simply would not bother to
search for it). Faced with such nonlinear differential equ-
ations the contemporary graduate student would automa-
tically rush to the computer to solve them by means of
numerical techniques.
This consistency between Newton’s geometric method
and my independent force balance method further
strengthens the position that the centrifugal force is a
real and useful force and not a fictitious one to be
avoided. What is really needed, though, is a comparison
between theory and measurements. That is not likely to
happen with most of the hypothetical orbital problems.
Perhaps the elliptical problem, with the force center at
the center of the ellipse, may lead to such a comparison
someday by means of a mechanical model (i.e. a mass
rotating and vibrating on a spring, see Section 6).
Could one use the same force balance method with the
centrifugal force to attack the famous unsolved 3-body
problem of astronomy? This is a far more ambitious
theoretical project than any that occurred to me before
(or any that came after either).
5. THREE-BODY PROBLEM
At the end of the 1990s, I attempted to solve the cele-
brated 3-body problem in a plane (i.e. in two dimensions)
using my force balance method, including the centrifugal
force, and I got farther than I had expected to get on the
first try. However, the equations started to look a little
messy so I stopped. But the recent progress with New-
ton’s Principia, in which all five of the orbital problems
were found to be consistent with the force balance me-
thod, gave me the extra courage to try again. On a vaca-
tion trip to New Mexico I did not have any notes with
me, but nevertheless I sat down and drew a set of train-
gles connecting the three masses. In doing so I just hap-
pened to define two angles differently than before, and
this produced enough of a simplicity in the trigonometry
that it eventually led to an initial breakthrough, which
allowed me to proceed further than I had done previ-
ously. Without my books and papers I had to rederive
most of the necessary mathematical relations; a few I re-
membered but had to check. Four pages in the notebook
were enough to contain the derivation of the kernel of
the solution of the 3- and 4-body problems, and then the
generalization to the n-body problem was just a rela-
tively simple extension of the same procedure.
As it turns out now my original solution had errors in
it, which were pointed out by one peer reviewer in due
course, but these mistakes were corrected one by one
until the paper was finally accepted after about a two-
year review period. Of course I am very grateful to this
particular reviewer, whoever he or she is. The word “so-
lution” is not quite accurate, since what I have done so
far is to set up the differential equations to be solved for
the orbits of the gravitating masses. I have not actually
solved these equations and I am not the most likely per-
son to do so either. The equations are strongly nonlinear