Basic concepts of Mathematical Physics
My student sometime asks me a question, why should I know mathematics to understand the physics. So for that we have to know what physics is and what is mathematics, which came first physics or mathematics. Probably we all know that physics came first then mathematics. I am giving you an example, in ancient age when human being lived in the cave, and knew very little about science; first found that a heavy weight like a huge stone can be pushed easily over round shape trunk rather than over the rough ground. They tried the same thing with other shapes of trunk but it was not that much easy, then they discover that wheel can make the work easier. They found that a particular shape of wheel is most efficient, that is they tried to figure out the shape of wheel and invented the circle, since then probably start the geometry. So, what I mean to say is mathematics is the tool to know and understand the physics better. without mathematics physics is incomplete.
Physics is a subject of studying the natural phenomenon happening around us continuously. We can perceive the facts in more details with the mathematical explanation. Suppose a heater is being used for cooking. Without knowing the power capacity of heater we cannot use it for cooking a beef. So we have to first design the heater with particular resistance and current to generate sufficient watt, so that we can use it. So to design for a particular resistance and the current we need to know the relation of heat generation in circuit i.e. Q/t=i2R/J, so we need mathematical expression. We cannot design a lock gate of a dam, without knowing the pressure of the water at that point i.e. P=pgh. So mathematics is a tool without which we cannot measure the phenomena and without the measurement we cannot apply the physics in our engineering applications properly and precisely.
We can model every phenomenon of physics with the help of mathematics. That is why mathematical physics has become more useful in research aimed at studying and solving problems of physics mathematically. Mathematical physics covers a broad academic area by the mixing of pure mathematics and physics. I would say the phenomenon is physics and its representation is mathematics. Theoretical physics is more linked with experimental physics, where mathematical physics is some model of that phenomenon and on that model further guessing or exploration of the hidden truth of that phenomenon. That is why gradually started new fields of mathematics like applied mathematics and analytical mathematics, the branches related to analyzing and modeling the physical problems.
Importance of Mathematical Physics
While first course of physics we encounter the force, which is rate of change of momentum. While modeling the force in mathematics it becomes d(mv)/dt. While we get the mathematical model of force we start rethinking that only velocity change is not the cause of force, but with the constant velocity and variable mass also is a cause of force. Using this principle we have designed the rocket, variable mass conveyor system etc.
Wave is one of the most versatile physical phenomena where we can use a mathematical sinusoidal model to explain like interference, superposition of waves. At the same time we can use the same mathematical model in a.c. electrical circuit to calculate the electrical impedance using the same sinusoidal model of generated alternating emf, which is same like wave nature.
We have invented simple arithmetic's while we had seen the regular phenomenon like 2 kg salt and 3 kg potato together can balance 5 kg water. so 2+3=5, is a simple addition discovered by ancient man. Then we invented algebra when we encounter the simple physical quantities like pressure and volume varies inversely i.e. P~1/V, we put them into a relation PV=constant. So if P=1,V=8 then at pressure P=2, V would be definitely 4 if other factors remain constant.
So many solutions are possible of this equation for each value of pressure. Then on the path of science growth we saw that 5+5 is not always 10 for the certain quantities like force. If I apply 5N force on a stone from one side and my friend is also applying the same 5N force from other side, then stone will not move at all. But if we together apply the 5N force from same side the stone may start rolling. So there are some physical quantities where direction is also matters, this lead to invent the field of physics like vector and mathematical modeling of this phenomenon is vector algebra. So as the science progress, more and more complex mathematics was required to explain and model the physical phenomena. Like why should we design a table with shape cuboids, why not spherical, because it will roll over if it is spherical due to minimum torque required in this shape. How objects are designed considering ergonomics required complex geometrical modeling. Similarly Trigonometry can help calculating the distance of a celestial object like sun, moon etc.
Innovation starts progress rapidly for last 3 centuries, we have invented the more complex mathematical models like calculus for analyzing force, fluid flow, thermal conduction, wave phenomenon, electromagnetic behavior etc.
Without help of mathematical tools, these phenomenon were perhaps could not be explained and used properly and hence their applications. Some areas of physics are almost in the mathematical model so far, and still to explore and verify the truth. These are like quantum field theory, Quantum Relativistic Theories etc.
Mathematical physicists primarily expand physical theories. Because of the required level of mathematical rigour, these researchers often deal with questions that theoretical physicists have considered to already be solved.
However, they can sometimes show that the previous solution was incomplete, incorrect, or simply too naive. Issues about attempts to infer the second law of thermodynamics from statistical mechanics are examples.
So theoretical physicists many times elucidate the heuristic, intuitive, and approximate arguments. Mathematical physicists verify or reject the fact by mathematical proof. Arguably, rigorous mathematical physics is closer to mathematics, and theoretical physics is closer to physics. This is why mathematical physicists are often members of the mathematics department. The effort to put physical theories on a mathematically rigorous footing has inspired many mathematical developments.
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