Tabak J. Mathematics and the Laws of Nature: Developing the Language of Science
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150 MATHEMATICS AND THE LAWS OF NATURE
Noether’s discovery is that each conservation law is a state-
ment about symmetry. In geometry symmetry is an important
organizing principle. A visual image is symmetric about a line, for
example, if what lies on one side of the line is the mirror image of
what lies on the other. This is an idea with which most of us are
familiar. A frontal image of the human body, for example, is sym-
metric about the line that passes vertically through the center of
the face. Mathematicians have generalized this idea in a number of
ways. In mathematics there is symmetry with respect to a point, a
line, and a plane, as well as other types of mathematical symmetry.
Noether discovered that each conservation law corresponds to a
particular type of symmetry. In other words given a conservation
law we can find a symmetry associated with it. Alternatively, sym-
metries in the equations correspond to conservation laws. This
means that if we begin from the point of view of geometry—if we
identify a particular symmetry within the equations—then it is, in
theory, possible to determine a conservation law corresponding
to this symmetry. Geometry and conservation laws turn out to be
two sides of the same coin, and geometry was once again at the
forefront of humanity’s attempts to understand nature.
The details of Noether’s discovery require quite a bit of math-
ematics, but if we apply her ideas to the conservation of energy
we can get a feeling for the insight behind her discovery. To do
this it is helpful to think of time as a kind of line. We can imagine
moving (translating) ourselves either forward or backward along
the line. In order for the law of conservation of energy to be valid
there must exist a translational symmetry with respect to time. This
means that regardless of how we might imagine moving back and
forth through time, the amount of energy in an isolated system
must remain constant.
To see why this must be so, imagine a heat engine operating
between two bodies of different temperatures. As we run the
heat engine, the temperatures of the two bodies become more
and more alike. The closer the two temperatures become one to
another, the less work our engine produces per unit of heat. In
theory it is always possible to run our engine backward and use
it as a heat pump so that it restores the two thermal reservoirs to
New Insights into Conservation Laws 151
their original state. If time is translationally symmetric (or what is
the same thing: if the first law of thermodynamics is true), then
the best we can expect from this procedure is to “break even.” The
work we performed on our engine in restoring the thermal reser-
voirs equals the work obtained by running our engine in the first
place. Or, to put it more succinctly: “work in” equals “work out.”
But if, instead, translational symmetry fails, we can do better than
this—better in the sense that we can run an engine at a “profit.”
In this case we simply choose a time to restore the thermal res-
ervoirs when restoring the reservoirs requires less work than
we have already obtained in work from the engine. If this were
possible then at the end of the cycle our machine would have no
net effect other than the production of work. Such machines are
impossible to build: They violate the first law of thermodynamics.
Translational symmetry with respect to time is valid only if energy
is conserved, and energy is conserved only if translational symme-
try holds with respect to time. Noether showed that the geometric
property of symmetry is a central organizing principle of nature.
Sergey L’vovich Sobolev
One of the more remarkable mathematical facts about 19th-
century research that the solutions that were discovered to the
conservation laws of the era belonged to a very small class of func-
tions. Algebraic functions, trigonometric functions, exponential
and logarithmic functions, and combinations of these functions
were the most common solutions, but even if other lesser known
functions are included, it is still true that a great deal of progress
was made with a very small set of functions. Nature, it seemed, was
not just mathematical, it demonstrated a marked preference for the
types of functions that today’s students study in precalculus classes.
That changed during the 20th century when engineers and sci-
entists began to study phenomena that were very different from
those considered by earlier generations of scientists. Especially
important were discontinuous phenomena, an example of which
is the shock wave. When a jet or rocket breaks the sound barrier,
it produces a shock wave that we hear as a sonic boom. A sonic
152 MATHEMATICS AND THE LAWS OF NATURE
boom is often modeled as a
“surface of discontinuity.” In
other words, the mathematical
description of a sonic boom
is as a surface (of zero thick-
ness) propagating through
the atmosphere. In addition
to sonic booms, many other
phenomena can be modeled
as shock waves. Shock waves,
for example, can form in solids
and liquids as well as gases.
Shock waves have a number of
peculiar properties, but their
most distinctive characteris-
tic is that they are accurately
modeled as surfaces of discon-
tinuity.
To better appreciate what it
means to introduce a surface
of discontinuity into a medi-
um such as air or water, con-
sider temperature. As material
passes through a shock wave,
its temperature will change
discontinuously. This is often described as a jump in tempera-
ture. On one side of the shock, the temperature has one (cooler)
value and on the other side of the shock, the temperature has
one (warmer) value. There are no intermediate temperature
values connecting the higher temperature with the lower one.
By contrast, in most ordinary situations just the opposite is true:
Temperature varies continuously and smoothly from one point in
space to the next. Imagine, for example, placing one end of a long
metal rod in a fire and resting the other end of the rod on a block
of ice. The end in the fire will soon reach the temperature of the
fire, and the end on the block of ice will soon reach the tempera-
ture of the ice. But between the fire and the ice, the temperature of
Sergey L’vovich Sobolev. One of
Sobolev’s most important innovations
is that he discovered a way to model
rates of change without requiring
that the underlying functions have
tangents. As a result it became
possible to develop mathematical
models for a much wider variety of
phenomena.
(Gennadii Demidenko)
New Insights into Conservation Laws 153
the rod will change in such a way that given any temperature value
between red-hot and ice-cold, there will be a place on the rod that
attains that temperature. This is one way of thinking about what
it means for something—in this case, temperature—to change
continuously from point to point.
And it is more than temperature that changes discontinuously as
a shock wave propagates. Other material properties such as pres-
sure and density often have one value on one side of the shock
and a different value on the other side, and because the shock has
no thickness—or more precisely because the model of the shock
has no thickness—there are no intermediate values connecting
the higher and lower values. The value of each material property
simply jumps—either up or down—from one side of the shock to
the other.
Shock waves have always existed. Lightning, for example, pro-
duces a shock wave by the almost instantaneous heating of the
air, which causes a very rapid expansion of the air that produces
the shock. Explosions produce shock waves too. But for many
years, shock waves drew only limited scientific and engineering
attention. One reason is technical. Shock waves occur suddenly
and dissipate quickly and so measurements are difficult to obtain.
Without measurements, it is difficult to separate useful ideas from
useless ones, and so progress was slow. But it is also true that
mathematical models of shock waves require ideas and techniques
that are quite different from those used to study processes that
vary continuously.
To see what was new about the mathematical study of shock
waves, recall that conservation laws (laws of nature) are state-
ments about rates of change. Mathematically, conservation
laws are expressed as differential equations, statements about
relationships among known functions and the derivatives of
unknown functions. The goal is to use these relationships to
identify the unknown functions. Before the 20th century, the
principal tool used to investigate rates of change was calculus,
and calculus could be used to compute in either direction: Given
a function p(x), early investigators learned to compute the func-
tion p
˙
(x), which represented the rate of change of p(x) at each
154 MATHEMATICS AND THE LAWS OF NATURE
point x. (See chapter 5.) Going from p(x) to p
˙
(x) is called “tak-
ing the derivative,” and it is relatively easy to do. Conversely,
given information about p
˙
(x) and some additional information
they learned to compute p(x), at least in some simple cases. This
is relatively hard. Solving for p(x) given information about p
˙
(x)
and the necessary initial conditions is called solving a differential
equation. By way of example, recall the equation that relates the
rate of change of momentum, which is often represented by the
expression p
˙
(x), to the applied force, which we represent with
the symbol F(x). The equation is p
˙
(x) = F(x). Once the force has
been specified and the state of the system at some point in time
has been specified, the momentum can be computed by solving
the differential equation. (And often, the dependent variable—
in this case p—depends on more than one independent vari-
able. Differential equations that involve multiple independent
variables are called partial differential equations; differential
equations that involve a single independent variable are called
ordinary differential equations. We will not emphasize the dif-
ference.) Because the laws of nature are often expressed as dif-
ferential equations, finding solutions to differential equations
is essential—in fact, it is sometimes equivalent—to progress.
Early researchers always assumed that the solutions that they
sought were smoothly continuous, and the mathematical ideas
and techniques that they developed made essential use of this
assumption.
By contrast, early researchers into discontinuous processes were
faced with functions or initial conditions that incorporated jumps
or that led to the formation of jumps in the solution at a later
time. Consequently, these researchers were faced with some stark
choices. They could assume that the old laws of nature were not
valid for discontinuous processes and begin the search for new
laws. Or they could retain the old laws of nature and develop new
and more applicable mathematical ideas and techniques, ideas
and techniques that took into account the notion of discontinu-
ous change.
They chose to retain the conservation laws that had proved to
be so useful and to expand the mathematics used to express and
New Insights into Conservation Laws 155
analyze them. It is easy to see why: Conservation laws are usually
more fundamental than the mathematics used to express them.
Think back, for example, to Newton’s first law of motion: “Every
body continues in its state of rest, or in uniform motion in a right
line, unless it is compelled to change that state by forces impressed
upon it.” Or Clausius’s formulation of the first law of thermo-
dynamics: “The energy of the universe is constant.” To be sure
these statements have mathematical formulations, but they are
not mathematical in themselves. And, as mentioned previously, the
same laws can have different mathematical formulations depend-
ing on the situation the researcher is trying to describe.
One of the first 20th-century mathematicians to generalize clas-
sical mathematical ideas in order to accommodate the new types of
problems was the Russian mathematician Sergey L’vovich Sobolev
(1908–89). Biographical descriptions of Sobolev concentrate
almost entirely on his mathematical accomplishments: He demon-
strated mathematical talent at an early age, and throughout his life
he devoted himself to the development of his mathematical ideas
and to the development of science and mathematics in the Soviet
Union. His dedication to mathematics was extraordinary. During
World War II as the invading German military approached
Moscow, Sobolev responded by moving his research activities
from Moscow to Kazan. A few years later, with the German army
in retreat, Sobolev returned to Moscow. These were some of the
most turbulent times in history. Many millions of Russians died
during World War II and millions more at the hands of their own
government under the leadership of Joseph Stalin, but nothing
distracted Sobolev. Today he is recognized as one of the most pro-
ductive and influential mathematicians of his time.
Sobolev’s best-known contributions are in three areas. He
helped to invent a branch of mathematics called functional
analysis. In the study of functional analysis, mathematicians cre-
ate “spaces” in which the “points” are individual functions. Each
such space has its own geometry, and operations on a space can
be interpreted as operations on classes of functions. The study
of these spaces has revealed a great deal of new information
about functions, certain types of equations, and the operators
156 MATHEMATICS AND THE LAWS OF NATURE
that transform one class of functions into another. Some of the
most important classes of function spaces are known as Sobolev
spaces. A second area in which Sobolev made a unique contri-
bution was in the study of cubature formulas, which involves
the calculation of volumes. Sobolev helped to extend common
formulas to higher dimensions. (By way of example, it is well
known that the area of a disc of radius r is
π
r
2
, and the volume of
sphere of radius r is 4
π
r
3
/3. Sobolev helped extend these types of
formulas to four and more dimensions—leading to the study of
higher dimensional spheres, higher dimensional ellipsoids, and
so forth.) His third contribution, and the one of most relevance
to this chapter, is Sobolev’s concept of derivative.
Recall that the idea of derivative is central to calculus. Leibniz
and Newton, for example, envisioned a derivative at a point on a
smooth curve as the slope of the (unique) line that is tangent to
the curve at the point. If, for example, the derivative to a smooth
curve at a point is a large positive number, it means that the curve
is rapidly increasing in the neighborhood of the point in question.
A large negative value for the derivative means that the curve is
rapidly decreasing in a neighborhood of the point. Smaller val-
ues meant less rapid rates of change. This concept of derivative
proved to be very fruitful. Generations of mathematicians were
kept busy exploring this idea and finding new uses for it, but by
the 20th century mathematicians, engineers, and scientists were
increasingly preoccupied with the study of processes that could
not be represented by smooth curves. Some of these curves had
corners; some had jumps; and some had more peculiar properties.
Some of the more “pathological” curves cannot be drawn even
in theory. In particular, these curves failed to have tangents at
specific points; some of the curves failed to have tangents at every
point. Consequently, rates of change, as the term was understood
by Leibniz and Newton, could not be associated with these curves.
But because of their importance, researchers still needed to study
the curves. Sobolev’s solution to this challenge was his concept of
generalized derivative.
As in other chapters of this book where other concepts such
as momentum and mass are discussed, it is important to bear in
New Insights into Conservation Laws 157
mind that there is more than one way to represent the “rate of
change” of a dependent variable. Newton and Leibniz discovered
a definition for the rate of change, but other definitions exist.
Just as scientists confused Newton’s model of the universe with
the universe itself, mathematicians confused the classical deriva-
tive with the phenomena it was meant to describe. Sobolev was
one of the first to realize that rates of change can be modeled
in ways that are fundamentally different than that discovered by
Newton and Leibniz. One of Sobolev’s great accomplishments
was to develop a new way of understanding and solving differen-
tial equations.
The definition of derivative in use in the 19th century, which is
the same definition that is used in calculus classes today, is appeal-
ing because it has an easy-to-appreciate geometric interpretation.
But the real test of a definition rests on whether it leads to useful
results—not whether it is easy to appreciate. Sobolev demon-
strated that his definition allowed him to solve equations that no
one had solved before. It took some time for many researchers
to catch up. Some researchers would use generalized derivatives
when there was no other choice but preferred to use classical solu-
tions whenever they were available. Generalized solutions were
not always perceived as “real” solutions. Today, however, Sobolev’s
work is part of mainstream mathematical research, and general-
ized solutions are recognized as just as valid as classical solutions.
Sobolev’s ideas have been accepted because without them prog-
ress in some areas is just not possible. Conservation laws are still
central to contemporary thinking, and one can show that classical
solutions do not exist for certain differential equations arising
from conservation laws. One must, therefore, adopt alternative
methods and alternative concepts of what it means to solve these
equations if they are to be solved at all.
Do these new types of solutions make sense? This can only be
addressed on a case-by-case basis as researchers compare their
generalized solutions with experiment. If agreement is good, that
is a compelling reason to continue to develop the ideas and tech-
niques for finding generalized solutions. Sobolev’s ideas continue
to attract the attention of researchers the world over.
158 MATHEMATICS AND THE LAWS OF NATURE
Olga Oleinik
Sobolev was just one of many distinguished Soviet mathematicians
of the time. The Soviet government vigorously supported research
into higher mathematics and established numerous research cen-
ters throughout the country dedicated to the study of advanced
mathematics. One of Sobolev’s most prominent intellectual suc-
cessors was the Soviet mathematician Olga Arsenievna Oleinik
(1925–2001). Born in Kiev, she received a Ph.D. from Moscow
State University. Her thesis adviser, a man of whom she always
spoke highly, was Ivan G. Petrovsky (1901–73), an internationally
recognized expert in the field of differential equations. Oleinik
taught at Moscow State University and eventually became head of
the department of differential equations. She wrote more than 300
research papers and a number
of books. She is remembered
as a prolific and highly creative
mathematician.
Oleinik applied Sobolev’s
ideas about generalized solu-
tions to partial differential
equations to obtain new insights
into conservation laws. The
paper for which she is best
remembered in this regard is
entitled “Discontinuous Solu-
tions to Non-Linear Differ-
ential Equations.” The word
“Discontinuous” in the title
refers to shocklike phenomena,
and the “Non-Linear” refers to
the form of the equation. Dif-
ferential equations are usually
classified as being either linear
or nonlinear. Without going
into the details, the nonlinear
equations are always harder to
solve than the linear ones.
Olga Arsenievna Oleinik successfully
applied new concepts in mathematics
and an abstraction of the idea of
entropy to the mathematical study of
discontinuous processes such as shock
waves. Her ideas continue to influence
the direction of mathematical
research.
(Gregory A. Chechkin)
New Insights into Conservation Laws 159
In her article, she demonstrated a number of deep insights into
the nature of conservation laws, but these are not the conservation
laws previously considered in this volume. Strictly speaking they
are not conservation laws at all if by a conservation law one means
a statement about an actual physical property. Instead, Oleinik
studied generalizations of conservation laws, abstract equations
that retain the general mathematical form of conservation laws. To
see what is new in this approach, recall that earlier chapters of this
book describe some ways that scientists, engineers, and mathema-
ticians have created models of nature. Oleinik, by contrast, stud-
ied models of models of nature. She attempted to identify general
patterns shared by all conservation laws, and she was especially
interested in conservation laws with discontinuous solutions. She
coupled her abstract approach with a generalized entropy condi-
tion (described later in this section) to produce results that were
new, useful, and profoundly insightful. Given the highly abstract
nature of her work, some words of introduction are necessary.
First, it should be emphasized that not all researchers have
found the second law of thermodynamics to be especially useful.
Recall that the second law of thermodynamics can be expressed in
terms of the property called entropy (see the sidebar “Entropy” in
chapter 7) and, unlike other laws of nature that can be expressed
as differential equations, the second law of thermodynamics is
expressed as a differential inequality, a mathematical expression
involving known functions, the derivatives of unknown functions,
and an inequality. Differential equations tend to be more useful
than differential inequalities because equalities are stronger math-
ematical statements. Equalities place tighter restrictions on the set
of possible solutions. Differential inequalities are weaker in the
sense that many possible solutions to an inequality exist, and there
is often no way to choose a single unique solution from among
them. And the most famous of all differential inequalities is the
mathematical formulation of the second law of thermodynamics,
which states that the entropy of an isolated system cannot decrease
over time—or to put it another way, the derivative of the entropy
for an isolated system is always greater than or equal to zero. (Or
to put it still another way, the graph of the entropy function of an