As a student and later as a teacher of both physics and mathematics, I have always been intrigued by the fact that some of the most important laws of physics are formulated by second order linear differential (SOLD) equations.
For readers not familiar with calculus or differential equations it suffices to know that SOLD equations are complicated equations but can still be solved exactly by analytical means and results expressed by formulae; anything more complicated would require use of a computer and results would be just sets of numbers which can be tabulated or presented as graphs.
In some cases, even though the laws did not say anything specifically about the linearity (i.e. proportionality), the physicists simplified the equations by neglecting non-linear terms; for example, by neglecting terms in frictional or other damping forces which might be dependent on the square of the velocity.
In other cases, there are individual laws representable by first order equations, but they can be made more complete and self-consistent by combining with other first order equations and converting them into SOLD equations. The best example are Maxwell’s equations obtained by combining four separate laws. Even Schrodinger’s equation in quantum mechanics is a SOLD equation.
This simplicity had some major benefits. First of all, it meant that these equations could be analytically solved and behaviour of various physical systems could be predicted with very precise accuracy under almost any given condition.
Secondly, one could combine these mathematical versions of laws of physics with other equations representing different physical conditions and derive other “laws” or corollaries. For example, one can show that Kepler’s “laws” in astronomy (which probably took Kepler and his associates years to come up with by empirical means) can be derived from Newton’s laws of motion in a straightforward way.
Understanding the entire phenomenon of electromagnetic waves and all its effects was a consequence of solving Maxwell’s equations. Thirdly, because of the analytical nature of the results one could design experiments and compare the predicted results with the measurements over a rather wide range of circumstances in order to in fact validate the laws themselves.
To emphasise these points let us consider the possibility that laws of physics had necessitated use of third (or higher) order and/or non-linear differential equations. For example, suppose Newton had discovered that applied force is not proportional to acceleration but square of the velocity or some function of the rate of change of acceleration.
What would have been the consequences? At the very least the evolution of physics would have been slowed down considerably and perhaps would have never reached the level of sophistication and completeness that we now take for granted.
Of course, all differential equations can be solved numerically using computers but computers are relatively new inventions. What would have physicists like Newton and Maxwell done? Presumably one could have generated some numerical solution even without a computer by developing a basic algorithm and the results would appear in the forms of long tables of numbers; after all Newton invented some of the earliest schemes of numerically solving differential equations.
These tables would have definitely showed some general trends such as how the velocity of an object changes with time under the application of some force, but no one would have any idea about the precise mathematical nature of such variations.
It would also be impossible to know if the behaviour described by numerical solutions obtained only applied for a given condition (specified by numerical values of some parameters) or valid more generally unless hundreds and perhaps thousands of these solutions were obtained numerically covering a wide range of possibilities.
Even if we compared these numerical results with experimental measurements we could not conclude anything about the validity of the laws because such comparison could only be performed for a finite limited set of circumstances and could not be generalized. We would not even know if some proposed law was actually a law. We would have been stuck with a set of postulates or educated guesses.
So was this ability to represent the fundamental laws of nature by SOLD just a lucky break for us or does it perhaps signify something very profound? One possible inference that can be drawn is that nature is inherently simple and is describable by relatively simple mathematical equations. This immediately casts doubts on the justification for all kinds of very sophisticated and difficult mathematical approaches in dealing with cosmology and behaviour of elementary particles.
Another more spiritual explanation is that God is just playing with us! Let me elaborate on this point with an analogy. You can keep your children confined to a large playroom and supply them with all kinds of simple educational toys like Lego sets, jigsaw puzzles and various building blocks.
These toys can be chosen in order to make sure that the kids can handle them and also learn from playing with them. The children study these toys and almost intuitively learn how to put them together apparently getting a huge satisfaction in the process.
So much so that they do it repeatedly and even try to apply this skill set to other things outside the playroom. They might even offer their help to their father who is perhaps busy with assembling some appliance or furniture. Of course, the parents would get a kick out of their children’s enthusiasm fully knowing that they have a very long way to go before they learn what goes on in the real world.
We are all like God’s children confined to this Planet Earth which is nothing more than a playground where we are supposed to learn some of the basics and not the whole truth. Because we are confined here we have access to only the objects and phenomena on the earth and we have developed a very systematic formulation of how things are built and work based on our intelligence and intuition.
We don’t realize that perhaps God deliberately made these objects and behaviour relatively simple so that we can understand them using our given intellect. Most of the physicists are arrogant enough to believe that they fully understand how all physical objects in the whole universe interact with each other.
Their confidence in their formulation is so strong that they would like to apply those to everything well outside the reach of this earth such as other galaxies and inside the quarks at the other extremity. However, all these efforts are perhaps silly just like a child trying to help his/her father using their skill learned from Lego pieces and Jigsaw puzzle.
Just like the children who imagine a world around them based on the fairytales they read in their playroom perhaps all the concepts in cosmology and all the theories about how the universe was created are nothing more than some fantasies! Is it not only surprising but also highly improbable that in spite of the existence of billions of stars in the universe our earth seems to be the only planet with an environment capable of supporting intelligent life who dare to explain it all?
I do not know about you, but my conviction is that we have just learned a simplified version of the real truth that applies only to everything around us on this earth. Physicists are clever people; they would introduce some fancy concept like “dark matter” that no one can see in order to explain apparent discrepancies. I am waiting for my death to reach the next level of knowledge base and start learning the absolute truth!
The writer, a physicist who worked in academia and industry, is a Bengali settled in America.