MM Research Preprints 53-59
KLMM AMSS Amdenha Sinica
Vol. 1, beb 198? 53
of
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]. 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 to 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 to 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) indeterminates X I , . . . ,
X , over the basic d-field F we shall associate
[ t c r d ] , called the INDEX-SET of P, in notation ind ( P ) , t = number of actual terms in P , a
P (f
4-tuple of integers c = the greatest subscript c for which X , occurs actually in P , to 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 to 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 deg(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 Ld, 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 , to be denoted by sep(P). Naturally, all these terminologies come from works of Ritt. ordering and
For d-pols in indeterminates
(<
X I , . defined in the following way. Let
[ t z c2 r2 dz ] resp. We say then Pi
. .
<<
, X ,
Pz over the d-field
Pi, Pz
F we shall consider a partial be d-pols with index sets if one of the following cases occurs: ci r1 d1 ]
(4 c1 < cz,
(b) c1 =
CZ, but
7-1
<
7-2,
295

54 Wu Wen-tsun
(c) c1 =
CZ, T I
= rz, but d l
< dp.
With respect to such a partial ordering of d-pols we can then introduce the notions of DIF-
FERENTIAL 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,”.) z n ( s T ) G =
3 k
Q k p k
+
R. in which I,, S, are the respective initials and separants of d-pols in d-ASC, L, certain non-negative integers which will be taken to be as small as possible, and Qk,
(abbr. d- REMDER) of G w.r.t. d-ASC, to be denoted as d-remdr(G/d-ASC). and M3 are
R d- pols with R d-reduced w.r.t. d-ASC. The d-pol R is accordingly called the d-REMAINDER
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 (DPS/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), d-zero (DPS) = d-zero (DCHRIK)
+ d-zero (DPSk). k
(11)
(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 to 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 ) (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 to 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 to the tube, of semi riiiiji)r mis of tlie elliptic: orl)it,s. from the sun to the planet.
In order to deduce mechanically the Newton’s laws
(K3) (actually only ( K i ) (Kz)
(Nz)
(K1) - 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
(K1) 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’
= h, (4) h’ = 0. ( 5 )
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:
+
(y”)’] = k , (6) k’ = 0, xy” = yx”.
(Nil 7)
( N z , 8 )
Now between the polar and the rectangular coordinates we have also the equations (9)
-
(13) below:
X =
T C O S W ,
(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

56 Wu Wen-tsun 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,
T , 2 ,
Y , 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) with Pi given by (1’)-(13’) as shown below:
(13) will turn to be the equations P, = 0
+1*
2 3 1
- 1
*
2 3 1
*
2 2 2
*
2 4 2
- 1
*
2 2 1 , (1’)
+1 * x i , ,
+1
*
2‘22,
+1
*
*.&I
- 1
*
251,
(20
(39
(40
+I
*
2431
*
( 2 g 2 ) 2
+1*
2151,
+
1
* *
(2y3)’ -
1
*
2 5 2 ,
+1
*
2152,
+1
*
2 3 2
*
2 6 3
- 1
*
2 3 3
* zgz,
+1
*
2 3 1
*
242 -
1
*
2 3 2 ,
+1
*
2 3 1
*
2 4 3
- 1
*
2 3 3 ,
+ 1 *
+1
* xi2
+
1
*
2f -
1,
+
1
*
2 4 3
*
.&I,
+I
* zk3
-
1
*
2 4 2
* xil.
(5’)
( 6 9
(7’)
(87
(90
(lo/)
(11’)
(129
(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
2 2 1
= p
2 3 1
= r
# 0 ,
2 2 2
= e
#
0,
# 0 ,
2 3 3 = 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 3 1
*
2;1
*
2 5 1
-1
*
2&
*
( 2 b 1 ) 2
+1
*
2’22,
+
1
*
2 3 1
*
2 2 1
*
( 2 b 1 ) 2 f
+
2
*
2z1
*
2 2 1
*
+1*
2z1
* xg2
*
2g1
- 1
*
2 3 1
2g1
+
*
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:
1 3 = -1
I4 = +1
*
5 3 1
*
2;1
*
2 2 2 , ect..
+
2
* *
5 2 1
+
1
*
221
*
2&
- 1
*
2:1 = +1
*
2 3 1
*
2&
* x i 3 ,
The separants are essentially the same as the initials, with at most a further factor of
2 3 3 .
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 case (14). The degenerate case for which one of
~ 2 1 , ~ 2 2 , ~ 3 1 , ~ 3 3 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’ -
= 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

58 Wu Wen-tsun
+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
+
’ ’ ’
-1
*
2 5 2 .
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,
-2
-1
$1
+1
* xi1 *
(2h1)2
*
2‘22,
+
1
*
2;l
*
2g1
*
5321
*
5 2 1
* +
1
*
2 3 1
* z ; ~
+
’ ’ ’
*
21311
+
. .
* 2i1 * & * &
- 1
*
2 3 1
*
2 2 1
*
(2’31)2,
+1
*
231 -
1
*
222
*
2 3 2
- 1
*
2 2 1 ,
+1*
2g2
+
1
* &
- 1
* &,
+I *
232
* x i 3 - 1
*
2 3 3
* x i z - 1
*
2 5 1 ,
+1
*
2,$
*
(&)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 to 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 =
+
( 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 to 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 the Laws of Motion to be found. The hypothesis d-polset is now consisting of
7 pols below:
H i
= +1
Hz = +l
*
212
-
*
2‘21,
1
*
2 2 2
*
231
- 1
*
Z Z ~ ,
H3
= +1
* x ~ Z ,
H4
H5
+ 1 * x : ~
Hs = +1
*
H7
=
= +1
*
2 3 1
+
*
X L ~ ,
= +1
* (ZS~)’
11;
+
2322
- 1
*
232
*
2 5 1
- 1
*
251,
1
*
- x T Z ,
( ~ $ 2 ) ~
*
211.
In a CPU-time of about 21 min., we find the final d-char-set to be consisting of 7 d-pols of which the first two d-pols are one in
2 1 1
= a’ alone and the other in
2 1 2
=
T and
211 = a’.
The 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 the process there appears a d-pol in the 4-th d-polset given by:
B = +4
* x;Z
*
211
+
1
*
5 1 2
*
B y our general principle of i.e.,
MTD B = 0 should be a consequence of Kepler’s Laws. The equation a consequence of the original d-polset,
B = 0 is however nothing else but the
Newton’s inverse square law ? * a = const.. We have thus discovered in a n automatic manner the Newton’s Law (N1) the d-pol from the Kepler’s Laws by means of our general Principle. Moreover,
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 the successive d-pols
Hz,
H3,
H I , H4, H5.
Hence we have also automatically discovered during the 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 Lab- oratory (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.,
(1986) 221-252.
4 (1984) 207-235. Republished in J. of Automated Reasoning, 2
[6] Wu Wen-tsun, A constructive theory of differential algebraic geometry. Preprint, to be published in Proc. DD6- Symposium in 1985 at Shanghai.
301