Theoretical physicists today still use a core technology
that was developed in the 18th century out of the calculus pioneered
by Isaac Newton and Gottfried von Leibniz.
Isaac Newton derived his three Laws of Motion through
close, almost obsessive observation and experimentation, as well as
mathematical reasoning. The relationship he discovered between force
and acceleration, which he expressed in his own arcane notation of fluxions,
has had the most impact on the world in the differential notation used
by his professional rival, Wilhelm von Leibniz, as the familiar differential
equation from freshman physics:
After Newton accused Leibniz of plagiarism in the
discovery of calculus, Leibniz' vastly more convenient and intuitive
differential and integral notation failed to become popular in England,
and so the majority of advances in the development of calculus in the
next century took place in France and Germany.
At the University of Basel, the multitalented Leonhard
Euler began to develop the calculus of variations
that was to become the most important tool in the tool kit of the theoretical
physicist. The calculus of variations was useful for finding curves
that were the maximal or minimal length given some set of conditions.
JosephLouis Lagrange took Euler's results and applied
them to Newtonian mechanics. The general principle that emerged from
the work of Euler and Lagrange is now called the Principle
of Least Action, which could be called the core technology of
modern theoretical physics.
In the Principal of Least Action, the differential
equations of motion of a given physical system are derived by minimizing
the action of the system in question.
For a finite system of objects, the action S
is an integral over time of a function called the Lagrange
function or Lagrangian L(q, dq/dt),
which depends on the set of generalized coordinates and velocities (q,
dq/dt) of the system in question.
The differential equations that describe the motion
of the system are found by demanding that the action be at its minimum
(or maximum) value, where the functional differential of the action
vanishes:
This condition gives rise to the EulerLagrange equations
which, when applied to the Lagrangian of the system in question, gives
the equations of motion for the system.
As an example, take the system of a single massive
particle with space coordinate x (in zero gravity). The Lagrangian is
just the kinetic energy, and the action is the energy integrated over
time:
The EulerLagrange equations that minimize the action
just reproduce Newton's equation of motion for a free particle with
no external forces:
The set of mathematical methods described above are
collectively known as the Lagrangian formalism
of mechanics. In 1834, Dublin mathematician William Rowan Hamilton
applied his work on characteristic functions in optics to Newtonian
mechanics, and what is now called the Hamiltonian
formalism of mechanics was born.
The idea that Hamilton borrowed from optics was the
concept of a function whose value remains constant along any path in
the configuration space of the system, unless the final and initial
points are varied. This function in mechanics is now called the Hamiltonian
and represents the total energy of the system. The Hamiltonian formalism
is related to the Lagrangian formalism by a transformation, called a
Legendre transformation, from coordinates
and velocities (q, dq/dt) to coordinates and momenta (q,p):
The equations of motions are derived from the Hamiltonian
through the Hamiltonian equivalent of the EulerLagrange equations:
For a massive particle in zero gravity moving in
one dimension, the Hamiltonian is just the kinetic energy, which in
terms of momentum, not velocity, is just:
If the coordinate q is just the position of the particle along the
x axis then the equations of motion become:
which is equivalent to the answer derived from the Lagrangian formalism.
Classical mechanics would have had a brief history
if only the motion of finite objects such as cannonballs and planets
could be studied. But the Lagrangian formalism and the method of differential
equations proved well adaptable to the study of continuous media, including
the flows of fluids and vibrations of continuous ndimensional objects
such as onedimensional strings and twodimensional membranes.
The Lagrangian formalism is extended to continuous
systems by the use of a Lagrangian density
integrated over time and the Ddimensional spatial volume of the system,
instead of a Lagrange function integrated just over time. The generalized
coordinates q are now the fields q(x) distributed over space, and we
have made a transition from classical mechanics to classical
field theory. The action is now written:
Here the coordinate x^{a} refers to both time and space, and
repetition implies a sum over all D+1 dimensions of space and time.
For continuous media the EulerLagrange equations
become
with functional differentiation of the Lagrange density replacing ordinary
differentiation of the Lagrange function.
What is the meaning of the abstract symbol q(x)? This
type of function in physics that depends on space and time is called
a field, and the physics of fields is called, of course, field theory.
The first important classical
field theory was Newton's Law of Gravitation, where the gravitational
force between two particles of masses m_{1} and m_{2}
can be written as:
The gravitation force F can be seen as deriving from
a gravitational field G, which if we set x_{1}=0
and x_{2}=x,
can be written as:
Newton's Law of Gravitation was the beginning of
classical field theory. But the greatest achievement of classical field
theory came 200 years later and gave birth to the modern era of telecommunications.
Physicists and mathematicians in the 19th century
were intensely occupied with understanding electricity and magnetism. In
the late 19th century, James Clerk Maxwell found unified equations of
motion of the electric and magnetic fields, now known as Maxwell's
equations. The Maxwell equations in the absence of any charges
or currents are:
Maxwell discovered that there exist electromagnetic
traveling wave solutions to these equations, which can be rewritten
as
and in 1873 he postulated that these electromagnetic waves solved the
ongoing question as to the nature of light.
The greatest year in classical field theory came in
1884 when Heinrich Hertz generated and studied the first radio waves
in his laboratory. Hertz confirmed Maxwell's prediction and changed
the world, and physics, forever.
Maxwell's theoretical unification of electricity and
magnetism was engineered into the modern human power to communicate
across space at the speed of light. This was a stunning and powerful
achievement for theporetical physics, one that shaped the face of coming
20th century as the century of global telecommunications.
But this was just the beginning. In the century that
was just arriving, the power of theoretical physics would grow to question
the very nature of reality, space and time, and the technological consequences
would be even bigger.
