MM Research Preprints 53-59
KLMM AMSS Amdenha Sinica
Vol. 1 , beb 198?
53
Mechanical Derivation of Newton’s
Gravitational Laws from Kepler’s Laws
W u Wen-tsun
Institute of Systems Science, Academia Sinica
It is an important historical event that Newton derived his laws from Kepler’s laws.
However, how the former ones can be deduced from latter ones is rarely touched upon in
current texts on calculus or mechanics, though the deduction of the latter ones from the
former ones is treated rather often in such texts, e.g. [l].The present preliminary report
aims at such a deduction, and, what is perhaps more important for our purposes, a deduction
in a MECHANICAL manner. The author owes for this report much t o Professor Gabriel of
Argonne National Laboratory. In fact, during a visit to Argonne in 1986 Prof. Gabriel told
the author such a problem for which he was already quite successful in applying his own
automated reasoning method based on works of Ritt can be applied as well t o deal with such
kind of problems.
To begin with, let us first recall some fundamental notions and the basic principles
underlying such method for which we refer for more details to [3,4] and [ 5 , 6 ] .
Let F be a DIFFERENTIAL FIELD (abbr. d-FIELD) which for the present paper may be
understood to be simply the d-field of all rational functions of some parameter t considered
as the time. To any DIFFERENTIAL POLYNOMIAL (abbr. d-POL) P (f0) in some
indeterminates X I , . . . , X , over the basic d-field F we shall associate a 4-tuple of integers
[ t c r d ] , called the INDEX-SET of P, in notation ind ( P ) ,as follows.
t = number of actual terms in P ,
c = the greatest subscript c for which X , occurs actually in P , t o be called the CLASS
of P , and be denoted as cls(P).
r = the highest order r for which the r-th derivative D,X, of the above X , occurs actually
in P, to be called the ORDER of P and t o be denoted as w d ( P ) .
d = the highest degree d of the above D,X, which occurs actually in P , to be called the
DEGREE of P and to be denoted by deg(P).
For a d-pol P with cls(P) = c, ord(P) = r , and d e g ( P ) = d, we shall calI the derivative
D,X, the LEAD of P , to be denoted by lead(P). Let L be this lead. Then the coefficient
of L d , which is itself a d-pol, is called the INITIAL of P , to be denoted as init(P). The
formal partial derivative of P w.r.t. L is then called the SEPARANT of P , t o be denoted
by s e p ( P ) . Naturally, all these terminologies come from works of Ritt.
For d-pols in indeterminates X I , . . . ,X , over the d-field F we shall consider a partial
ordering (< defined in the following way. Let Pi, Pz be d-pols with index sets [ t ici r1 d1 ]
and [ t z c2 r2 dz ] resp. We say then Pi << Pz if one of the following cases occurs:
(4 c1 < cz,
(b) c1 = CZ, but
7-1
< 7-2,
295
Wu Wen-tsun
54
(c) c1 = CZ,
TI
= rz, but d l
< dp.
With respect to such a partial ordering of d-pols we can then introduce the notions of DIFFERENTIAL ASCENDING SET, DIFFERENTIAL BASIC SET, and DIFFERENTIAL
CHARACTERISTIC SET ( abbr. d-ASC-SET, d-BAS-SET, and d-CHAR-SET resp. )just
as in the case of ordinary polynomial. We define also the notion of d-REDUCED as that of
REDUCED in the ordinary case.
Consider now a d-asc-set d-ASC consisting of d-pols
(d-ASC)
with
0 < cls(P1) < cZs(P2) < ’ ’ ’ < ClS(P,).
For any d-pol G we have then the following REMAINDER FORMULA:
n(I,”.)n ( s T ) G =
z
3
k
Qkpk
+ R.
in which I,, S, are the respective initials and separants of d-pols in d-ASC, L, and M3 are
certain non-negative integers which will be taken to be as small as possible, and Qk, R dpols with R d-reduced w.r.t. d-ASC. The d-pol R is accordingly called the d-REMAINDER
(abbr. d- REMDER) of G w.r.t. d-ASC, t o be denoted as d-remdr(G/d-ASC).
A finite set of non-zero d-pols is called a DIFFERENTIAL POLSET ( abbr. d-POLSET ).
Let such a d-polset DPS be given. A d-pol in the same indeterminates X , but over an
arbitrary DIFFERENTIAL EXTENSION FIELD (abbr. d-EXT-FIELD), F’ of F will be
said to be a SOLUTION (abbr. SOL) or d-ZERO of the set DPS if it satisfies all the
equations P = 0 for P in DPS. The totality of all such solutions or d-zeros will be denoted
by d-zero ( DPS ) and the totality of only those which are not d-zero of a given d-pol G will
be denoted by d-zero (D P S /G ).
Given a d-polset DPS we can deduce, just as in the ordinary case, a d-char-set DCHR in
a mechanical way. We have then, also as in the ordinary case, the formulas below:
d-zero (DPS) C d-zero (DCHR),
(1)
d-zero (DCHRIK) C d-zero (DPS),
(11)
d-zero (DPS) = d-zero (DCHRIK)
+
k
d-zero (DPSk).
(111)
in which K is the product of all initials and separants of d-pols in DCHR, and DPSk are
d-polsets which are the enlarged DPS with one of the initials or the separants adjoined t o it.
The formulas (I) (111) are at the basis of all our considerations about mechanization of
mathematics in the case involving differentiation.
Come now to the problem proper as cited in the title of the present paper. Let us first
formulate the Kepler’s laws ( K ) and the Newton’s laws (N) in the manner as given below:
~
(K1) The planets move in elliptic orbits around the sun as focus.
( K z ) The vector from the sun t o the planet sweeps equal areas in equal times.
296
Mechanical Derivation of Newton’s Laws
55
(K3)The
squares of periods of planet is motions are proportional t o the tube, of t h semi
mis of tlie elliptic: orl)it,s.
( N I )The acceleration of a planet is inversely proportional t o the square of the distance
from the sun to the planet.
( N z )The acceleration vectors of planets are directed toward the sun.
riiiiji)r
In order to deduce mechanically the Newton’s laws
(Nl),
(Nz)from Kepler’s laws (K1) -
(K3) (actually only ( K i )- (Kz)will be sufficient) let us take first coordinates and transform
the various laws into equation forms as follows.
Take polar coordinates with the sun at the pole and the major axis of the elliptic orbit
as the polar axis. Then the orbit will have an equation of the form
T
=p/(l -e
* cosw)
(1)
in which w is the angle between the polar axis and the vector from the sun to the planet. The
Kepler’s law ( K 1 ) corresponds to the equation (1) and also (2)-(3) below taken together:
p = const , or p’ = 0,
(2)
e = const , or e‘ = 0,
(3)
in which the prime means derivative w. r . t . the time t. Similarly Kepler’s law (Kz) will
correspond to the equations (4), ( 5 ) below:
T2W’
(4)
= h,
(5)
h’ = 0.
Let us take also rectangular coordinates (x,y) associated to the above polar coordinates
(T, w). Then the Newton’s laws N l , Nz will correspond to the following set of equations:
+
T~[(x”)’ (y”)’] = k ,
(6)
k’ = 0,
( N i l 7)
xy” = yx”.
( N z , 8)
Now between the polar and the rectangular coordinates we have also the equations (9) - (13)
below:
X = TCOSW,
(9)
y = T sin w ,
cos’w
+ sin’w
= 1,
(10)
(11)
(cosw)’ = -(sinw)w’,
(12)
(sinw)’ = +(cosw)w‘.
(13)
To proceed further let us first remark that it is immaterial whether the equations (9) - (13)
are dependent or not. What is important for us is that the computer can not recognize
any irrational or transcendental entities like sin w or cos w. This can however be remedied
297
Wu Wen-tsun
56
simply by treating cos w and sinw just like indeterminates connected by relations (11)- (13).
To apply our implemented programs let us now introduce indeterminates in replacing the
various functions by x’s as given below:
( P ,e , T
, 2 ,
Y , w , cosw, sinw, h , k) = ( ~ 2 1 r 2 2 2 ~ ~ 3 1 r 2 3 2 , 2 3 3 r 5 q 1 , 2 4 2 , 2 4 3 , 2 ~ 1 , 2 5 2 ) .
With this change of notations the equations (1)- (13) will turn to be the equations P, = 0
with Pi given by (1’)-(13’) as shown below:
+1* 2 3 1 - 1* 2 3 1 * 2 2 2
* 242 - 1* 221,
(1’)
+1 * x i , ,
(20
+1 * 2‘22,
(39
+1 *
*.&I
- 1 * 251,
(40
+1* 2151,
+I
* 2431 * ( 2 g 2 ) 2 + 1 *
(5’)
* (2y3)’
-
1 * 252,
+1 * 2152,
(69
(7’)
+1 * 2 3 2
* 2 6 3 - 1 * 2 3 3 * zgz,
+1 * 2 3 1 * 2 4 2 - 1* 2 3 2 ,
+1 * 2 3 1 * 2 4 3 - 1 * 2 3 3 ,
(lo/)
+ 1 * 2f
(11’)
+1*
+1 * xi2
+ 1*
-
(87
(90
1,
* .&I,
+I * zk3 - 1* 2 4 2 * xil.
(129
243
(13’)
Take now the d-polset DPS to consisting of the 11 d-pols (1’)-(6’), (9’)-(13’) of the
above set. Remark that the planets move in true non-degenerate elliptic orbits so that we
have
221
231
= p # 0 , 2 2 2 = e # 0,
=r # 0 , 233 = y # 0 .
In applying our algorithm for the finding of d-char-set DCHR of DPS we can then remove
any factors 2 2 1 , 2 2 2 , 2 3 1 and 2 3 3 during the procedure. The DCHR is found to be the 3-th
d-bas-set consisting of the 10 d-pols C, given below:
+1* 2’21,
+1 * 2’22,
-1
* 2 3 1 * 2;1 * 2 5 1 + 1 * 2 3 1 * 2 2 1 * ( 2 b 1 ) 2 f
-1 * 2& * ( 2 b 1 ) 2 + 2 * 2z1 * 2 2 1 * 2g1 +
+1* 2z1 * xg2 * 2g1 - 1 * 2 331 * 2g1,
298
’ ’ ’
‘’ ‘
Mechanical Derivation of Newton’s Laws
57
The CPU-time for bringing up this d-char-set is 146 sec.. The non-trivial initials are:
13
= -1
I4 = +1
* 5 3 1 * 2;1
* 222,
+2*
* 5 2 1 + 1 * 221 * 2& - 1* 2:1
= +1
* 2 3 1 * 2& * x i 3 ,
ect..
The separants are essentially the same as the initials, with at most a further factor of
233.
The proof of the Newton’s laws is now readily done. In fact, we find the d-remdrs of the
d-pols (7‘) and (8’) to be both 0 w.r.t. the above dpolset DCHR. By the equation (I) and
the remainder formula we see then the Newton’s laws are true at least in the non-degenerate
, ~zero
3 3 can be dealt with in
case (14). The degenerate case for which one of ~ 2 1 , ~ 2 2 , ~ 3 1 is
a similar but much easier way.
The Newton’s laws have thus been derived in a mechanical way from the Kepler’s laws as
required. However, in proving that the remainders are zero it requires, somewhat unexpected,
a quite long time, viz, a CPU-time of 10875.6 sec.. This defect comes seemingly from
two sources. One is due to inadequacy of programming in the procedure of reductions
so that improvement of the implementation of program is needed. A second one is due
to inadequate choice of coordinate systems. Thus, instead of a mixed use of polar and
rectangular coordinate systems we have tried to use the rectangular system alone. In this
way the Kepler’s law ( K 1 ) will correspond to following equations
together with equations (2) and (3). Similarly, the Kepler’s law (Kz) becomes the equation
zy’
- y2’
=h
(17)
with h satisfying (5). Replacing now the various functions by the x’s as before we have then
to consider a d-polset DPS’ consisting of 7 d-pols (2’), (3’), ( 5 ’ ) , (6’) and those corresponding
to (15)-(17), viz.
+1
* 2 3 1 - 1* 2 2 2 * 2 3 2 - 1 * 2 2 1 ,
+1*
2’21,
299
Wu Wen-tsun
58
+1*2‘22,
+1* 2&
+ 1* 2z3 - 1* 2z1,
+I * 532 * x i 3 - 1* 2 3 3 * xi2 - 1 * 5 5 1 ,
+1 * x;1,
+1 * xi1 * (
2 g
+ 1 * xi1 * (4,)3 +
’ ’ ’
* 252.
-1
The d-char-set DCHR’ is readily found in a CPU-time of 106.2 sec to be consisting of
the following 7 d-pols C: as the 2-th d- bas-set, viz.
+1 * 2‘21,
$1
* 2‘22,
+ 1* 2;l * 2g1 +
-2 * 5321 * 5 2 1 * Z ~+I 1 * 2 3 1 * z ; ~* 21311 + . .
-1 * 2i1 * & * & - 1 * 2 3 1 * 2 2 1 * (2’31)2,
+1 * xi1 *
+1 * 231
(2h1)2
-
’ ’ ’
1 * 222 * 2 3 2 - 1 * 2 2 1 ,
+1* 2g2
+ 1* & - 1* &,
+I * 2 3 2 * x i 3 - 1 * 2 3 3 * x i z - 1* 2 5 1 ,
+1* 2,$
* (&)2 + 1 * 2t1 * (2;3)2 +
-1 * 2 5 2 .
’ ’ ‘
,
The remainders of the d-pols (7’) and (8’) w. r. t. DCHR’ are again found t o be zero in a
shorter CPU-time of 5949.7 sec. The Newton’s laws are thus again deduced from the Kepler’s
laws in a mechanical way a little simpler than the way before. It seems that improvement
of the programming will further simplify the proofs in shortening the CPU-time to probably
less than half an hour. We remark that times are naturally all referred to the computer
which we are in use.
The proof presuppose that the Newton’s Laws are already known and require merely a
verification. Now suppose that we are in the stage of knowing the Kepler’s experimental
Laws alone, but entirely ignorant of what will be the form of the underlying Laws of Motion.
The Principle in the form of (I) - (111) now furnishes us a method of automatically discover
such unknown governing Laws. For this purpose let us introduce the acceleration a by
a2 = ( x ” )+
~ ( Y ” ) ~ arranging the order of the various entities involved in setting
(p,e,
2 ,
y,
r , h , a’) = ( 2 2 1 , ~ 2 2 , 5 3 1 , 2 3 2 , 2 1 2 , 2 5 1 , 2 1 1 ) .
Remark that we have deliberately arranged a and T t o be the first two indeterminates in
expecting to find some relation between them BS few first d-pols in the d-char-set which
300
Mechanical Derivation of Newton’s Laws
59
would give us t h e Laws of Motion to be found. T h e hypothesis d-polset is now consisting of
7 pols below:
= +1 * 212 - 1 * 2 2 2 * 231 - 1 * Z Z ~ ,
Hz = +l * 2‘21,
H3 = +1 * x ~ Z ,
H4 = + 1 * x : ~+ 11;2322 - 11; x T Z ,
H5 = +1 * 2 3 1 *
- 1 * 2 3 2 * 2 5 1 - 1 * 251,
Hs = +1 * X L ~ ,
2
H7 = +1 * (ZS~)’
1* ( ~ $ 2 )-~ 1 * 211.
Hi
+
In a CPU-time of about 21 min., we find t h e final d-char-set t o b e consisting of 7 d-pols of
which t h e first two d-pols are one in 2 1 1 = a’ alone and t h e other in 2 1 2 = T and 211 = a’.
T h e first one gives us thus a differential equation observed by the acceleration. This equation
and the second one between a and T are both too complicate t o be of any interest. However,
during t h e process there appears a d-pol in t h e 4-th d-polset given by:
B = +4 * x;Z
* 211 + 1 * 5 1 2 *
B y our general principle of MTD B = 0 should be a consequence of t h e original d-polset,
i.e., a consequence of Kepler’s Laws. T h e equation B = 0 is however nothing else but the
Newton’s inverse square law ? * a = const.. We have thus discovered in a n automatic manner
from t h e Kepler’s Laws by means of our general Principle. Moreover,
t h e Newton’s Law (N1)
the d-pol
H8 = H5
He = +l * 2 3 1 * Z;z - 1 * 2 3 2 * ~ $ 1
+
has its d-remainder already 0 w.r.t. the first d-bas-set BS1 consisting of t h e successive d-pols
H z , H3, H I , H4, H5. Hence we have also automatically discovered during t h e procedure the
theorem H8 = 0, i.e., Newton’s Law (Nz).
References
[l] Courant,R., Differential and integral calculus, vol 2 (1936).
[2] Gabriel, J.R., SARA- A small autometed reasoning assistant. Preprint, Argonne National Laboratory (1986).
[3] Ritt,J.F., Differential equations from the algebraic standpoint, Amer. Math. Spc., (1950).
[4] Ritt,J.F., Differential algebra, Amer. Math. SOC.,(1950).
151 Wu Wen-tsun, Basic principles of mechanical theorem-proving in elementary geometries,
J.Sys.Sci. 8~ Math. Scis., 4 (1984) 207-235. Republished in J. of Automated Reasoning, 2
(1986) 221-252.
[6] Wu Wen-tsun, A constructive theory of differential algebraic geometry. Preprint, to be published
in Proc. DD6- Symposium in 1985 at Shanghai.
301