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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