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đu; Ằ ` Gm¿m (Ui /)
mị th = TT TH, (9.18)
J=I 19
".. » _ CmunjG¡ —)
đt m Tỉ,
Further, we defne r¿¿ as the distance between the two planets ¿ and 7; this is
equal to
tụ = V8 — #7)? + (Mi — 9ý)” + (#¡ — 22). (9.19)
AIso, 3) means a sum over all values of j—all other bodies——except, of course,
for j ==. Thus all we have to do is to make more columns, /o2#s more columns.
W© need nine columns for the motions of Jupiter, nine for the motions of Saturn,
and so on. Then when we have all initial positions and velocities we can calculate
all the accelerations from Eq. (9.18) by first calculating all the distances, using
Eq. (9.19). How long will it take to do it? TỶ you do i9 at home, it will take a
very long timel But in modern times we have machines which do arithmetic
very rapidly; a very good computing machine may take 1 microsecond, that is, a
millionth of a second, to do an addition. To do a multiplication takes longer, say
10 microseconds. lt may be that in one cycle of calculation, depending on the
problem, we may have 30 multiplications, or something like that, so one cycle will
take 300 microseconds. 'Phat means that we can do 3000 cycles of computation
per second. In order to get an accuracy, of, say, one part in a billion, we would
need 4 x 105 eycles to correspond to one revolution of a planet around the sun.
That corresponds to a computation time of 130 seconds or about two minutes.
Thus it take only 6wo minutes to follow Jupiter around the sun, with all the
perturbations of all the planets correct to one part in a billion, by this methodl
(It turns out that the error varies about as the square of the interval e. lÝ we
make the interval a thousand times smaller, it is a million times more accurate.
So, let us make the interval 10,000 times smaller.)
So, as we said, we began this chapter not knowing how to calculate even
the motion of a mass on a spring. Now, armed with the tremendous power of
--- Trang 194 ---
Newton”s laws, we can not only calculate such simple motions but also, given
only a machine to handle the arithmetic, even the tremendously complex motions
of the planets, to as hipgh a degree of precision as we wishl
--- Trang 195 ---
I0
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10-1 Newton?s Third Law
On the basis of NÑewton”s second law of motion, which gives the relation
between the acceleration of any body and the force acting on it, any problem in
mmechanies can be solved in principle. EFor example, to determine the motion of
a few particles, one can use the numerical method developed in the preceding
chapter. But there are good reasons to make a further study of Newton”s laws.
First, there are quite simple cases of motion which can be analyzed not only
by numerical methods, but also by direct mathematical analysis. For example,
although we know that the acceleration of a falling body is 32 ft/sec2, and
trom this fact could calculate the motion by numerical methods, i% is much
casier and more satisfactory to analyze the motion and fñnd the general solution,
8 = 8g + 0g + 162. In the same way, although we can work out the positions of a
harmonic oscillator by numerical methods, ït is also possible to show analytically
that the general solution is a simple cosine function of £, and so it is unnecessary
to go to all that arithmetical trouble when there is a simple and more accurate
way to get the result. In the same manner, although the motion of one body
around the sun, determined by gravitation, can be calculated point by point by
the numerical methods of Chapter 9, which show the general shape of the orbit,
1E is nice also to get the exact shape, which analysis reveals as a perfect ellipse.
Unfortunately, there are really very few problems which can be solved exactly
by analysis. In the case of the harmonic oscillator, for example, if the spring
force is not proportional to the displacement, but is something more complicated,
one must fall back on the numerical method. Ôr ïf there are two bodies goïng
around the sun, so that the total number of bodies is three, then analysis cannot
produce a simple formula for the motion, and in practice the problem must
be done numerically. “That ¡is the famous three-body problem, which so long
challenged human powers of analysis; it is very interesting how long it took
people to appreciate the fact that perhaps the powers of mathematical analysis
--- Trang 196 ---
were limited and ¡it might be necessary to use the numerical methods. Today an
enormous number oŸ problems that cannot be done analytically are solved by
numerical methods, and the old three-body problem, which was supposed to be
so difficult, is solved as a matter of routine in exactly the same manner that was
described in the preceding chapter, namely, by doing enough arithmetic. However,
there are also situations where both methods fail: the simple problems we can
do by analysis, and the moderately difcult problems by numerical, arithmetical
methods, but the very complicated problems we cannot do by either method. A
complicated problem 1s, for example, the collision of two automobiles, or even
the motion of the molecules of a gas. There are countless particles in a cubic
millimeter of gas, and ¡it would be ridiculous to try to make calculations with
so many variables (about 10!——a hundred million billion). Anything like the
motion of the molecules or atoms of a gas or a block or iron, or the motion of
the stars in a globular cluster, instead of just two or three planets goïing around
the sun——such problems we cannot do directly, so we have to seek other means.
In the situations in which we cannot follow details, we need to know some
general properties, that is, general theorems or principles which are consequences
of Newton's laws. One of these is the principle oŸ conservation of energy, which
was discussed in Chapter 4. Another is the principle of conservation oŸ momentum,
the subject of this chapter. Another reason for studying mechanics further is
that there are certain patterns of motion that are repeated in many diferent
circumstances, so iÈ is good to study these patterns in one particular cireumstance.
For example, we shall study collisions; diferent kinds of collisions have much
in common. In the fow of Ñuids, it does not make mụuch diference what the
fuid is, the laws of the fow are similar. Other problems that we shall study
are vibrations and oscillations and, in particular, the peculiar phenomena of
mmechanical waves—sound, vibrations of rods, and so on.
In our discussion of NÑewton”s laws it was explained that these laws are a kind
of program that says “Pay attention to the forces,” and that Newton told us only
two things about the nature of forces. In the case of gravitation, he gave us the
complete law of the force. In the case of the very complicated forces between
atoms, he was not aware of the right laws for the forces; however, he discovered
one rule, one general property of forces, which is expressed in his 'Third Law, and
that is the total knowledge that Newton had about the nature of forces—the law