Whence the Force of F = ma?
When I was a student, the
subject that gave me the most trouble was classical mechanics. That
always struck me as peculiar, because I had no trouble learning more
advanced subjects, which were supposed to be harder. Now I think
I've figured it out. It was a case of culture shock. Coming from
mathematics, I was expecting an algorithm. Instead I encountered
something quite different— a sort of culture, in fact. Let me
Problems with F = ma
Newton's second law of motion, F = ma, is the soul of
classical mechanics. Like other souls, it is insubstantial. The
right−hand side is the product of two terms with profound meanings.
Acceleration is a purely kinematical concept, defined in terms of
space and time. Mass quite directly reflects basic measurable
properties of bodies (weights, recoil velocities). The left−hand
side, on the other hand, has no independent meaning. Yet clearly
Newton's second law is full of meaning, by the highest standard: It
proves itself useful in demanding situations. Splendid, unlikely
looking bridges, like the Erasmus Bridge (known as the Swan of
Rotterdam), do bear their loads; spacecraft do reach Saturn.
The paradox deepens when we consider force from the perspective
of modern physics. In fact, the concept of force is conspicuously
absent from our most advanced formulations of the basic laws. It
doesn't appear in Schrödinger's equation, or in any reasonable
formulation of quantum field theory, or in the foundations of
general relativity. Astute observers commented on this trend to
eliminate force even before the emergence of relativity and quantum
In his 1895 Dynamics, the prominent physicist Peter G.
Tait, who was a close friend and collaborator of Lord Kelvin and
James Clerk Maxwell, wrote
"In all methods and systems which involve the idea of force there
is a leaven of artificiality. . . . there is no necessity for the
introduction of the word "force" nor of the sense−suggested ideas on
which it was originally based."1
Particularly striking, since it is so characteristic and so
over−the−top, is what Bertrand Russell had to say in his 1925
popularization of relativity for serious intellectuals, The ABC
"If people were to learn to conceive the world in the new way,
without the old notion of "force," it would alter not only their
physical imagination, but probably also their morals and politics. .
. . In the Newtonian theory of the solar system, the sun seems like
a monarch whose behests the planets have to obey. In the Einsteinian
world there is more individualism and less government than in the
The 14th chapter of Russell's book is entitled "The Abolition of
If F = ma is formally empty, microscopically obscure, and
maybe even morally suspect, what's the source of its undeniable
The culture of force
To track that source down, let's consider how the formula gets
A popular class of problems specifies a force and asks about the
motion, or vice versa. These problems look like physics, but they
are exercises in differential equations and geometry, thinly
disguised. To make contact with physical reality, we have to make
assertions about the forces that actually occur in the world. All
kinds of assumptions get snuck in, often tacitly.
The zeroth law of motion, so basic to classical mechanics that
Newton did not spell it out explicitly, is that mass is conserved.
The mass of a body is supposed to be independent of its velocity and
of any forces imposed on it; also total mass is neither created nor
destroyed, but only redistributed, when bodies interact. Nowadays,
of course, we know that none of that is quite true.
Newton's third law states that for every action there's an equal
and opposite reaction. Also, we generally assume that forces do not
depend on velocity. Neither of those assumptions is quite true
either; for example, they fail for magnetic forces between charged
When most textbooks come to discuss angular momentum, they
introduce a fourth law, that forces between bodies are directed
along the line that connects them. It is introduced in order to
"prove" the conservation of angular momentum. But this fourth law
isn't true at all for molecular forces.
Other assumptions get introduced when we bring in forces of
constraint, and friction.
I won't belabor the point further. To anyone who reflects on it,
it soon becomes clear that F = ma by itself does not provide
an algorithm for constructing the mechanics of the world. The
equation is more like a common language, in which different useful
insights about the mechanics of the world can be expressed. To put
it another way, there is a whole culture involved in the
interpretation of the symbols. When we learn mechanics, we have to
see lots of worked examples to grasp properly what force really
means. It is not just a matter of building up skill by practice;
rather, we are imbibing a tacit culture of working assumptions.
Failure to appreciate this is what got me in trouble.
The historical development of mechanics reflected a similar
learning process. Isaac Newton scored his greatest and most complete
success in planetary astronomy, when he discovered that a single
force of quite a simple form dominates the story. His attempts to
describe the mechanics of extended bodies and fluids in the second
book of The Principia3
were path breaking but not definitive, and he hardly touched the
more practical side of mechanics. Later physicists and
mathematicians including notably Jean d'Alembert (constraint and
contact forces), Charles Coulomb (friction), and Leonhard Euler
(rigid, elastic, and fluid bodies) made fundamental contributions to
what we now comprehend in the culture of force.
Physical, psychological origins
Many of the insights embedded in the culture of force, as we've
seen, aren't completely correct. Moreover, what we now think are
more correct versions of the laws of physics won't fit into its
language easily, if at all. The situation begs for two probing
questions: How can this culture continue to flourish? Why did it
emerge in the first place?
For the behavior of matter, we now have extremely complete and
accurate laws that in principle cover the range of phenomena
addressed in classical mechanics and, of course, much more. Quantum
electrodynamics (QED) and quantum chromodynamics (QCD) provide the
basic laws for building up material bodies and the nongravitational
forces between them, and general relativity gives us a magnificent
account of gravity. Looking down from this exalted vantage point, we
can get a clear perspective on the territory and boundaries of the
culture of force.
Compared to earlier ideas, the modern theory of matter, which
really only emerged during the 20th century, is much more specific
and prescriptive. To put it plainly, you have much less freedom in
interpreting the symbols. The equations of QED and QCD form a closed
logical system: They inform you what bodies can be produced at the
same time as they prescribe their behavior; they govern your
measuring devices— and you, too!— thereby defining what questions
are well posed physically; and they provide answers to such
questions— or at least algorithms to arrive at the answers. (I'm
well aware that QED + QCD is not a complete theory of nature, and
that, in practice, we can't solve the equations very well.)
Paradoxically, there is much less interpretation, less culture
involved in the foundations of modern physics than in earlier, less
complete syntheses. The equations really do speak for themselves:
They are algorithmic.
By comparison to modern foundational physics, the culture of
force is vaguely defined, limited in scope, and approximate.
Nevertheless it survives the competition, and continues to flourish,
for one overwhelmingly good reason: It is much easier to work with.
We really do not want to be picking our way through a vast Hilbert
space, regularizing and renormalizing ultraviolet divergences as we
go, then analytically continuing Euclidean Green's functions defined
by a limiting procedure, . . . working to discover nuclei that
clothe themselves with electrons to make atoms that bind together to
make solids, . . . all to describe the collision of two billiard
balls. That would be lunacy similar in spirit to, but worse than,
trying to do computer graphics from scratch, in machine code,
without the benefit of an operating system. The analogy seems apt:
Force is a flexible construct in a high−level language, which, by
shielding us from irrelevant details, allows us to do elaborate
applications relatively painlessly.
Why is it possible to encapsulate the complicated deep structure
of matter? The answer is that matter ordinarily relaxes to a stable
internal state, with high energetic or entropic barriers to
excitation of all but a few degrees of freedom. We can focus our
attention on those few effective degrees of freedom; the rest just
supply the stage for the actors.
While force itself does not appear in the foundational equations
of modern physics, energy and momentum certainly do, and force is
very closely related to them: Roughly speaking, it's the space
derivative of the former and the time derivative of the latter (and
F = ma just states the consistency of those definitions!). So
the concept of force is not quite so far removed from modern
foundations as Tait and Russell insinuate: It may be gratuitous, but
it is not bizarre. Without changing the content of classical
mechanics, we can cast it in Lagrangian terms, wherein force no
longer appears as a primary concept. But that's really a
technicality; the deeper questions remains: What aspects of
fundamentals does the culture of force reflect? What
approximations lead to it?
Some kind of approximate, truncated description of the dynamics
of matter is both desirable and feasible because it is easier to use
and focuses on the relevant. To explain the rough validity and
origin of specific concepts and idealizations that constitute the
culture of force, however, we must consider their detailed content.
A proper answer, like the culture of force itself, must be both
complicated and open−ended. The molecular explanation of friction is
still very much a research topic, for example. I'll discuss some of
the simpler aspects, addressing the issues raised above, in my next
column, before drawing some larger conclusions.
Here I conclude with some remarks on the psychological question,
why force was— and usually still is— introduced in the foundations
of mechanics, when from a logical point of view energy would serve
at least equally well, and arguably better. The fact that changes in
momentum— which correspond, by definition, to forces— are visible,
whereas changes in energy often are not, is certainly a major
factor. Another is that, as active participants in statics— for
example, when we hold up a weight— we definitely feel we are doing
something, even though no mechanical work is performed. Force is an
abstraction of this sensory experience of exertion. D'Alembert's
substitute, the virtual work done in response to small
displacements, is harder to relate to. (Though ironically it is a
sort of virtual work, continually made real, that explains our
exertions. When we hold a weight steady, individual muscle fibers
contract in response to feedback signals they get from spindles; the
spindles sense small displacements, which must get compensated
before they grow.4)
Similar reasons may explain why Newton used force. A big part of the
explanation for its continued use is no doubt (intellectual)
1. P. G. Tait, Dynamics, Adam & Charles
Black, London (1895).
2. B. Russell, The ABC of Relativity, 5th rev.
ed., Routledge, London (1997).
3. I. Newton, The Principia, I. B. Cohen, A.
Whitman, trans., U. of Calif. Press, Berkeley (1999).
4. S. Vogel, Prime Mover: A Natural History of
Muscle, Norton, New York (2001), p. 79.
Frank Wilczek is the Herman
Feshbach Professor of Physics at the Massachusetts Institute of
Technology in Cambridge.
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