Appendix to Chapter 2
Newton’s Laws
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A1. In the latter part of the 17th century, Isaac Newton, building on
the work of many predecessors, formulated a small number of laws from which
quantitative predictions about the movements of objects in the heavens can be
made. It was soon realized that some
movements of terrestrial objects could also be predicted with Newton's laws.
While celestial objects are nowadays seen to change, and even in a
certain sense to be born, live and die, the Newtonian laws according to which
they change seem to be permanent, although they have been extended in various
ways. Newton's laws and the
myriad of consequences which have been drawn from them make up classical
or Newtonian mechanics, sometimes called rational or analytical
mechanics. The part of classical
mechanics which applies to the motions of objects in the heavens is commonly
called celestial mechanics.
A2. In his textbook on classical mechanics (1985), Laurence Taff
observes that classical mechanics rests on Newton's three Laws of Motion, and
he states them as they are in Newton's Principia, 1687 (translated from
Latin):
Newton's
First Law. "Every
body continues in its state of
Newton's
Second Law. "The
change of motion is proportional
The
motion of a body is defined by Newton to be the product of a quantity
called the mass of the body, which measures its reluctance to change
its state, with the velocity of the body, which measures the rate at
which its distance from some reference point is changing, and also specifies a
direction in which the change is taking place.
This is called momentum today.
The velocity and/or direction may change at each instant of time.
The change in motion is actually the rate of change of
momentum. Except in a few simple
cases a quantitative statement that this rate of change of momentum is
proportional to impressed forces requires the techniques of the mathematical
discipline known as calculus. To
say the rate of change of momentum is proportional to
the impressed forces is to say that it is some fixed number multiplied by the
quantity which measures the force at each point of space and instant of time.
The particular fixed number or constant to be used is different
for different units of measurement for time, distances and forces (second or
years, meters or feet or miles, pounds or dynes, etc.). Often impressed forces are different at each point of space,
but at any one given point are the same for each instant of time.
Newton's Second Law is the most dominant of the three laws of motion
since it gives a recipe for forming differential equations.
These are statements made using concepts of calculus.
In many cases they can be solved using methods of calculus, in one or
another sense of the word solved (including approximate solutions), to
give quantitative descriptions of the behavior of a great number of physical,
chemical, biological, geological, statistical, and other kinds of systems.
It can be shown that the first law can be derived as the special case
of the second law in which the magnitude of the impressed forces is zero.
When the word motion in the second law is interpreted as momentum,
and this meaning is used in the first law, the statement in the first law that
a body tends to continue in a state of uniform motion in a straight line can
be interpreted to mean that the momentum of a body in such a state will stay
the same as it moves, so Newton's First Law contains a law of conservation
of linear momentum.
Newton's
Third Law. "To every
action there is always opposed
This
should not be taken to mean that objects never move. If I push on you, thus exerting a force, and you move
backwards, an explanation according to Newton’s Third Law is that your
reaction push was at the instant of contact equal in magnitude to my push,
though in the opposite direction (along a straight line). This diminished the
magnitude of my push in an amount equal to the magnitude of the push you
exerted. However, although my
push was weakened, there was still some more of my push it left over, so to
speak, so you were subjected to an acceleration in the direction of my push
–-- and you moved.
To do celestial mechanics, Taff observes, one supplements these
Newton's
Law of Universal Gravitation. "Every
particle in the
Thus
the force of gravity exerted by one particle on another particle
can be measured by finding numbers measuring their masses in some way, and
multiplying these together; then finding the distance between the particles in
some way and squaring it and dividing the result into the product of the
masses; and finally, multiplying by a fixed number determined by the units of
measurement being used. (Laurence
G. Taff, Celestial Mechanics, A Computational Guide for the Practitioner,
1985, p. 1-2; Taff's quotations from Newton's Principia are from the
translation by Florian Cajori, 1934, p. 13-14, and Newton's definitions of motion,
mass (or quantity of matter and vis insita), impressed
force, etc., are given on p. 1-6.) The
gravitational forces which bodies exert on other bodies are
determined by regarding bodies as made up of particles in some way, and
using techniques of calculus. This
is not a very easy task, on the whole. Its
study is known as potential theory (for reasons we won't go into here).
A3. Having stated these laws of classical mechanics, and supplemented
it with Newton's Law of Gravitation in order to do celestial mechanics, Taff
observes that there is essentially no more physics in his book -- the rest is
mathematics. In effect, Taff defines classical mechanics to consist
of the consequences of Newton's three laws of motion, as worked out using
methods of mathematics, and celestial mechanics to consist of the
consequences of the laws of motion together with the law of gravitation.
From this point of view, the "impressed forces" spoken of in
Newton's Second Law are confined to gravitational forces when doing what one
might call “pure” celestial mechanics.
A4. The word mechanistic
is open to conflicting interpretations. Some
have taken it to be opposed to animistic, so a mechanistic
universe is one in which planets and the like have no internal principles of
change, as they did for Aristotle and countless others.
In particular, for some, divine guidance is precluded in a mechanistic
universe. The attitude is
captured in a story about Laplace. Napoleon
is supposed to have asked Laplace why he never mentioned the Creator in his
work on celestial mechanics, and Laplace is supposed to have replied:
"Sire, j'ai pu me passer de cette hypothèse" -- "Sir, I
have been able to dispense with that hypothesis."
A5. Others have taken a mechanistic universe to be one made out
of gear wheels, pulleys, levers, springs and the like, in the manner of a machine,
which runs and has run forever on its own.
However, the author of the Laws of Motion, Newton, believed that a
Creator was involved in the working of the world. Aside from divine guidance, he also speaks in Definition III
of the Principia of bodies having inertia or vis insita (innate
force), an internal power of resisting change in motion, tending to
make it continue in whatever state it is in.
This attributes to machines something beyond their mere extension in
space and time. Because of this
proposal, and because he was not able to find a satisfactory mechanical model
for his theory of gravity (although he made a few conjectures), Newton was
accused by followers of Descartes of introducing so-called "occult
powers" into natural philosophy of the kind which had been popular among
medieval scholastic philosophers, and which Descartes had been at great pains
to banish. Descartes himself had
tried to base a theory of gravity on the motion of vortices -- little
whirlpools, so to speak. An
important part of Newton's purpose in writing his Principia was to show
that Descartes's model doesn't work as an explanation of gravitation.
A6. Thus the connection of
classical mechanics with machines is not as close as some have thought.
There is also the question of mathematics. E. J. Diksterhuis made a study of the transition to
"classical science" which took place during the 17th century, and
came to this conclusion: "The
mechanization of the world-picture during the transition from ancient to
classical science meant the introduction of a description of nature with the
aid of the mathematical concepts of classical mechanics; it marks the
beginning of the mathematization of science, which continues at an
ever-increasing pace in the twentieth century."
(E. J.