Mechanics is the first topic to be studied in most, if not
all, high-school physics courses. Also, many students think
that it is the easiest to understand. Indeed, historically
it was the first branch of physics to be made vigorous three
centuries ago, in the Principia published by Newton, who got
rid of the false notions of Aristotle and planted physics
as the first tree in the garden of sciences.
(Picture of Isaac Newton - This
photo was downloaded from American Institute of Physics website
http://www.aip.org/)
What, then, is mechanics exactly? Has it become such a time-honored
subject that nothing new can really come out of it, and after
teaching its basis to students we may just put it aside?
The second question depends on the answer to the first, so
to begin with we'll clarify what mechanics studies. It may
be interesting to note that I have asked this question before
university undergraduates in Cambridge, Okazaki (Japan) and
Hong Kong. In the former two places some gave satisfactory
replies, but in the last, none, so far. What about you?
My answer is as follow. In mechanics we examine certain kinds
of motion of objects. The object may be an electron, a piece
of rock, water in a river, an athlete, ... anything. However,
only some, but not all, types of motion executed by an object
fall within the scope of mechanics, namely, those involving
explicitly only the mass of the object and no other properties
of the object.
Let me elaborate. An object may have many properties ("attribute"),
e.g., electrical charge, but in mechanics we do not concern
ourselves with equations of motion that contain charges as
parameters. In fact, they form the subject matter for electrodynamics,
another branch of physics. We do study the oscillation of
an object attached to an elastic spring, although the elastic
force is electrical in nature, being the sum total expression
of the chemical bonds among molecules in the spring. However,
the charges of the molecules need not, and cannot, be explicitly
taken into account in Hooke's Law.
Other examples of attributes excluded from mechanics are
the "colors" of certain elementary particles (studied
in chromodynamics, or the mechanics for nuclear and high energy
physics), and the "free will" of an athlete (studied
in psychology).
Thus mechanics has a narrow theme. But its realm is gigantic.
Exclusion of electromagnetic forces appears to be a severe
restriction, because these forces are often so much stronger
than say friction or viscosity under comparable circumstances.
However, precisely because of this strength of electromagnetic
interaction, an object is usually neutral in electrical charges
and the effects of its positive and its negative charges almost
cancel out, as is the case of a solid body sliding over a
solid surface, where the residue of the sum total of electrical
forces among the molecules of the two solids is called friction.
In face, mechanics is applicable to many common situations.
It suffices for grand occasions too!
Having defined mechanics, we now examine what new twigs
have sprouted from such an old branch. In the early part of
the 20th Century, Schrodinger, Heisenberg, Dirac and others
helped to develop quantum mechanics, which explains the world
of small objects more accurately than Newtonian ("classical")
mechanics. At about the same time, Einstein almost single-handedly
worked out General Relativity, which is a theory of gravity,
in which the gravitational mass of an object is strictly proportional
to its inertial mass (in this way mechanics studies motions
that really involve no independent attribute of the object!)
(Picture of Erwin Schrodinger
- This photo was downloaded from American Institute of Physics
website http://www.aip.org/)
(Picture of Werner Heisenberg
- This photo was downloaded from American Institute of Physics
website http://www.aip.org/)
(Picture of Paul Adrien Maurice
Dirac - This photo was downloaded from American Institute
of Physics website http://www.aip.org/)
With the extension to relativity, mechanics fulfils its grandest
scope, which is to describe the large-scale structure of the
universe, with the exception of the first few minutes in the
beginning of the universe, where forces depending on charges
and colors were as important as forces depending on masses.
I would encourage some of the more motivated students to learn
a little bit of General Relativity. The investment will be
so tiny compared to what you will then be able to comprehend.
The whole universe!
Back on earth, significant progress continues in the subject
of mechanics. One of the most notable advances in recent years
pertains to chaotic dynamical systems, which became recognized
as such in the 1970's and remain under intense study today.
Before closing our survey on what lies beyond high-school
mechanics, therefore, we shall take a look at chaos.
Of course, we have just proclaimed the greatness of mechanics.
Indeed, in as early a time as in the beginning, Newton himself
calculated how the earth's rotational axis precessed (as slowly
as 0.7
per century) due to the non-spherical shape of the earth (which
is bigger at its equator than at its poles, but only by 0.3%).
Today, too, we can send a spacecraft off to visit other planets
in succession over a period of many years, without significant
deviations from the calculated trajectory and with only minor
orbital adjustments en-route. In these types of problems in
mechanic, once the positions and velocities of all the relevant
objects (earth, moon, sun etc.) are known at one time, we
can determine their values at any other time. The solar system
is a "deterministic" mechanical system.
However, can the Hong Kong Observatory predict to the same
accuracy how strongly will the wind blow and in what direction,
after a time as short as an hour? No, at least not with mechanics!
The reason is that no one knows the positions and velocities
of all the relevant objects at even one time, because these
objects, namely the molecules in the air around Hong Kong,
are exceedingly large in number. The atmosphere is a "statistical"
mechanical system. We may calculate its properties only statistically,
not because the air molecules follow some new laws of mechanics,
but because there are far too many of them.
Consider, lastly, a simple coin dropped vertically onto
a flat surface. Can you predict which side faces up afterwards?
The coin is simple in the sense that it is rigid and ideally
symmetrical in shape and constitution, but after its fall
one side is up and the other down, so what force breaks the
symmetry of the situation as it hits the surface?
The state of a rigid body may be described by merely the
positions and velocities of any three points, therefore the
randomness of the outcome cannot be blamed on any statistical
nature of the drop. On the other hand, the outcome does depend
extremely sensitively on the initial conditions: a tiny tilt
in one direction or another just before impact will cause
the coin to flip over! In this case, a tiny deviation in the
trajectory of the moving body will lead to a disproportionally
large divergence within a short time, and this is the defining
characteristic of a "chaotic" dynamical system.
Chaos is one reason why a system does not exhibit the high
degree of symmetry that the underlying principles of organization
possess. This is why there are so many interesting patterns
in the real world, despite the "simplicity" of the
laws of physics, chemistry, biology, ... maybe even sociology.
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