Since 1687, when Newton published his famous three laws of motion incorporating all the features of motion in a mathematically consistent form, a lot has been
accomplished in physics. Developments in physics have grown exponentially
within last hundred years and continues to do so. The plethora of volumes in
different branches of the field is enough to intimidate any student and in-fact
this is one of the prime reasons that students give up careers in physics at a very
early stage. A problem that every physics student faces is how to cover such a
vast literature which includes fields of such diverse variety as classical
and quantum mechanics, electrodynamics, thermal and
statistical physics, nuclear and particle physics, atomic and molecular
physics, solid state physics, field theories, etc. Each
one of these subjects contains so much that it seems it might take a lifetime
to master even a single of them. Moreover, the degree of mathematical
intricacies involved only seem to make the situation worse.
Let us consider the case of Newtonian Classical Mechanics. What we have here are Newton's three laws of motion. The principles of (mechanical) energy and momentum conservation are consequences of these three laws which appear mathematically for a certain type of forces (known as conservative forces). These three laws along with Newton’s inverse square law of gravitation forms the content of the whole Newtonian classical mechanics and enable us to analyze a wide variety of mechanical systems including the planetary system. Knowing these laws we should be able to solve any problem in the realm of classical mechanics. But now if we ask ourselves the following question :
Given these laws, can I show that
the planets move in an elliptical trajectory
and derive all the relevant
quantities like angular momentum, energy, etc. of
the planetary system?
I am confident
that the majority of people capable of doing this will be very very small
though most of us have been learning these laws since our high-school or
freshman year (junior college). Using the same laws, we can analyse systems
like a rotating top, we can deduce Bernoulli's theorem, motion of a rocket,
kinetic theory of gases, colliding systems, rigid bodies, etc.
Similarly, if we know the Newton’s three laws, Coulumb's law, the conservation of charges, the fact that moving charges give rise to a current and the Biot-Savart law, then in principle we know the whole of classical electrodynamics. Everything in classical electrodynamics is governed by these laws. But despite knowing all these (which is sufficient for analysing any system involving charges), we have a hard time to crack problems based on electrostatics and magnetostatics. So where do we lack?
The difficulty
comes in with the mathematics. It is here that we need to realize the true
meaning of the phrase Physics is a mathematical theory of nature and I’ve tried to do this through some of
my previous posts.
Importance of mathematics in physical sciences can
hardly be over-emphasized. Mathematics provides an unambiguous and concise way
of expressing the laws or ideas. Once an idea is expressed in mathematical
form, you can use the rules (axioms, theorems, etc.) of mathematics to change
it into other statements. If the original statement is correct, and you follow
the rules correctly, your final statement will also be correct. This is what we do when we
"solve" a mathematics problem – we convert a statement into another
mathematically equivalent statement. For instance,
(a) the
statement : “net force acting on a body is zero” is mathematically equivalent
to “momentum of the body is conserved.”
(b) the
statement : “force acting on a system is path independent” is mathematically
equivalint to “total mechanical energy of the system is conserved.”
(c) the
statement : “potential of a force field is spherically symmetric” is equivalent
to “angular momentum of an object in this force field is conserved”, etc.
Each of the
above statements can be converted into other by using rules of mathematics. The
advantage is that mathematics helps bring out relations between quantities that
are not directly obvious from experimental observation. For example, it is not
obvious that the angular momentum of an object should be conserved in a
spherically symmetric potential. But it can be arrived at through simple
mathematics using Newton’s laws.
As I already
pointed out earlier, every branch of physics is actually based on a certain
small set of fundamental laws. These laws, in their mathematical form, are concise,
exact and universal statements formed by carefully examining the observations
from a large number of experiments. They contain a lot of information and have
far reaching consequences. I have tried to demonstrate this with the concept of
inertial frames of reference – how it follows from the very definition of
inertial frames that the velocity of light should be a constant and how the Newton’s
first law is justified.
But it is only when we treat these laws in a proper
mathematical way that the things unfold and we are able to comprehend their
consequences. It is all about treating these laws mathematically to get the information
we want. The knowledge of laws that we require to do physics is really small,
but it is the experience of using mathematics that we lack, which makes the
subject a hard nut to crack.
From the
importance of mathematics follows our next question. How to know what kind of
mathematics should we learn for physics? This is a question that haunts many (I
think every) physics students and why should it not, mathematics, being the
most advanced science, contains ginormous amount of matter. At times, it can be
overwhelming for students. This is where the importance of problem solving
comes in the picture. As physics (or science) students, our job is to solve
problem based on natural or artificial phenomena. We first make observations,
then state/define the problem and then proceed on to solving it. If we are not
able to solve it, we try to change our approach and look for other suitable
mathematical methods (which usually alredy exist in mathematics). Thus, it is
while solving the problems that we learn mathematics and if the problem is
fundamentally different from the existing ones, then the solution becomes a
physical theory. This is how the physics that we know today came into being –
by attempts of people trying to solve problems or explain their observation.
This is how the calculus was born.
The mathematics
learnt this way is relevant as well as useful. Regular problem solving practice
helps us to get conversant with the appropriate mathematics required and build
our own set of mathematical techniques. But the most important is that it helps
in developing a kind of mathematical intuition which enables one to quickly
understand the problem and make accurate and judicious guesses. The more we
practice, the more this intuition gets closer to reality and better we get in
understanding more intricate phenomena. This ability is a great asset and marks
experts from the mediocre or second-rate scientists. This is what all of us (as
science students) strive for knowingly or unknowingly.
Another advantage
of learning mathematics this way is that we begin to realize that there is a
large overlap between the mathematics we require to work in different branches
of physics. For example working knowledge of Fourier transforms, enables us to
solve a wide variety of differential equations in mechanics, electrodynamics,
statistical physics. In optics, the Fourier transforms provide a great way to
express superposition of waves and is a powerful tool to deal with diffraction
and interference problems. The same Fourier transforms are extensively used in
Quantum mechanics to express and relate quantities like position and momentum
and in a way plays a key role in probabilistic interpretation of wave-functions.
Just learning Fourier transform technique enables us to work on so diverse
problems. And the same is true for all other mathematical techniques. They are
universal and their uses are many-folds. Thus problem solving helps us realize
the connections between different branches of physics (actually the science)
and helps in understanding nature with a broad perspective. It is obvious that
a person with such an in-depth mathematical (and intuitive) understanding is
likely to perform better as a scientist than those who specialize in particular
fields (and are called specialists which most of us are). This is another key
difference between original thinkers and mediocre thinkers.
Therefore, the better way of learning physics would be reading less and solving more. Solve
as many problems as you can and from as many fields as you can. The best way to
do this is - take a bunch of plain papers, a pen and a book of problems on
physics and start solving by yourself (don’t take a peek on answers). It may
happen that you may not be able to solve a particular problem (or most of the
problems) immediately, you may require a few minutes, a few hours, or a few
days, but that’s okay, you must have the vigor,
doggedness, determination, perseverance to continue with your efforts. You have
to go an extra mile to stand out of the crowd and to be able to do something
novel. There is really no substitute of hard work here, there are no
shortcuts and it is necessary to do so to stay competitive and sharp. We must
train ourselves in such a way that given a problem at any time on any subject we should be able to solve it or at least be able guess how to approach the
problem. We usually already have enough knowledge of laws that we require to solve
classical problems, but what we need is to TEACH OURSELVES how to use that
knowledge through mathematics. That is what we (physics students) spend most of
our life-time on : learning how to use what we already know.
A disturbing trend that I have observed among young students is that they acquire the notion very early in their career that they are going to specialize in a particular branch of physics and hence they need to be conversant with things concerning that particular branch only. NEVER ever make the mistake of thinking that you will be working on a particular field in physics and need to concentrate only on that. There is a reason why we are called students of physics and not students of classical mechanics or students of quantum mechanics or students of astrophysics or students of material science. What we do is physics and we should be able to do it at any place, any time. We MUST be equally good in all of them and this requires a lot of practice. To get a feel of what I mean here read the biographies of people like Feynman, Landau Curie, Kelvin, Maxwell, etc. who made significant contributions to physics. Although you may know them for their contributions to a particular branch but they all were universalists.
To summarize, my advice would be to solve as many problems as you can. All theories that exist today are the results of efforts of people trying to explain/solve some kind of problem. It always goes from problems to theory, never backwards. Unfortunately we are always taught the subject the other way round in our schools and hence all the unnecessary problems. The subject is vast in applications but really small in content.
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