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What would you choose as the Top 5 Equations? Top 10?

Equations are of central importance in all of science and engineering, but especially so in mechanics.

Even leaving aside algebraic equations, handbooks on PDEs alone list hundreds of equations. However, a few of these do stand out, either because they encapsulate some fundamental aspect of physics/science/engg., or because they serve as simpler prototypes for more complex situtations, or simply because they are so complex as to be fascinating by themselves. There might be other considerations too... But the fact is, some equations really do stand out as compared to others.

If so, what equations would you single out as being most important or interesting? To make the matters more interesting, first, please think of making a short list of only 5 equations. Then, if necessary, make it one of 10 equations---but no more, please! :)

As to me, here is my list, put together in a completely off-hand manner:

Top five:
(1-3) The linear wave-, diffusion- and potential-equations.
(4) The Schrodinger equation
(5) The Navier-Stokes equation

Additionally, perhaps, these equations:
(6) The Maxwell Equations
(7) The equation defining the Fourier transform
(8) Newton's second law (dp/dt = F)
(9) The Lame equation (of elasticity)

Am I already nearing the limit or what... Hmm... But, nope, I am not sure whether I want to include E = mc^2. ... I will give this entire matter a second thought some time later on.

But, how about you? What would be your choices for the top 5/10 equations? Why?


PS: Also posted today in the Computational Scientists & Engineers group at LinkedIn, and also will post soon at my personal blog.




Zhigang Suo's picture

Here are two equations that might make my choice of top 10 equations:

  • S=logW
  • dU=TdS

It is hard to search for greatness in the equations themselves.  Like any other great equations, they symbolize deep and far-reaching ideas.


Nzaoui's picture

ZAOUI Noureddine
Electromechnical Engineer


(1) Gauss equations

(2) Maxwell equations

(3)  The equation defining the Fourier transform

(4)  E = mc^2

(5) not yet unification equation,,,


absolutely totally :)

SivaSrinivasKolukula's picture

I prefere the following:

1. Newton's Second law

2. Conservation of Energy

3. Conservation of linear momentum

4. Conservation of angular momentum

      All physics lies with in those laws and to derive every equation we begin from the above laws/equations... 

Hi all,


0. Thanks for all your suggestions. (I lost my first draft while typing---accidentally closed the reply window. Will try to recapture as much thought as possible.) Detailed comments below.


1. Zhigang, I completely missed out on these two equations! Thanks for introducing a principle as fundamental and as general as the second law of thermodynamics, in this discussion!!

Out of those two equations, however, methinks, dS = dU/T would always compete better than S = k ln W to make it in the shortest of the short lists. And there go I---I now have to face the wrath of so many physicists!! But still...

The application of certain basic conceptual ideas concerning the nature of the physical universe (which, I, in my notes on "The Universe: Finite or Infinite" (still under preparation) call the principle of composition), together with the more basic continuum definition of entropy, could lead to the Boltzmann equation. However, theoretically tracing the converse would be difficult, IMHO. That's the general idea why I would pick out dS = dU/T over S = k ln W.


2. Noureddine, I don't really expect the so called unifying equation to be here any time soon, or, for that matter, ever. The nature of physics is just not like that. Still, if it is possible to have such an equation, then every other equation would be encompassed by it. Surely, also e = mc^2, which, in any case, is only an implication of the classical EM theory together with Lorentz transformation. 

Now, the energy-mass conversion was a profound insight, theoretically, and of a lot of consequence, practically (think atomic energy and atom bombs). Yet, while compiling the shortest of the short list of equations of all times, I am not sure if it merits to be mentioned indpendently, regardless of what hippies, music- and "culture"-people in general have made of it.

Similarly, the Gauss equation is already a part of the Maxwell equations. So, it need not be mentioned separately.

But then, you still have one (or six) open slots anyway! What would you fill them with?


3.  "sasaborg", I think the Euler identity can be said to have been already well-absobed in the Fourier transform equation (the way we know it, in its modern form---i.e. including the cosines as well as the sines.)

I am curious what you would have to say for the remaining 4 (or 9) slots which are, like, totally like, still open to you!


4. Siva, a couple of points:

4.1 I think that you can perhaps collapse the two momenta conservation principles into one. If you do that, then, with Newton's second law (defining the force and the torque) as an additional given, the energy conservation principle already comes out as an entailment---it need not be stated separately.

However, though personally I am not too sure of it, the mainstream physicists' view seems to be that the converse is not necessarily true. They say that the momentum conservation always holds---whether in the relativistic regime or the quantum mechanical one. But not the energy conservation.

4.2 Which brings me to the second point. Not all of physics lies within those four laws; you have to specify some additional equations/ideas to take you to the relativistic and quantum mechanical regimes. 

Anyway, you still have one (or six) more slots open for you to think of!!


5. Finally, how about a few more equations that we could perhaps be talking about?

How about Hiesenberg's relation (Dp Dx = \hbar/2)? Here, again, many physicist suggest that the parallel relation concerning energy and time (De Dt =\hbar/2) is not on the same footing. (That was one consideration behind my point 4.1 here.)

How about the Fokker-Planck equation? the d'Alembertian form of wave equation? the Dirac equation?

Any others?


6. I have been talking about this business of collapsing or subsuming equations into more abstract equations. A word is in order.

It is sometimes said that you can take all conservation principles as nothing but mere applications of Noether's principle. Thus, if you cast Noether's principle into symbolic form, i.e. in the form of an equation, then all conservation equations can be said to be automatically deducible.

However, to me, this line of thought seems uncalled for. Noether's principle seems to be rather a formal rearrangement of the already known physics rather than any *new* physical observation/discovery standing on its own ground. That's why I would be in favor of a more detailed listing of the equations---they seem to capture the physics of it better.

Further, physics is an empirical science. If, tomorrow, some new physical phenomenon gets discovered, it is possible that the integration then required would be such that the more abstract schemes such as Noether's theorem might have to be jettisoned, but *some* of the more detailed equations might continue to hold. For instance, it is difficult to see how the second law might at all get violated!

This point, together with a constrain as artificial as having to have only five/ten/fifteen equations, is what makes the compilation of such lists so very interesting.


A long reply, once again! But, yes, I do look forward to more short-lists!! It's interesting, isn't it? (Thanks in advance.)


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