I'm reading book where I realized that in many math books, i.e. Calculus, ODE, DG etc., many theorems and propositions were formalized in terms of first or second derivatives, or just any arbitrary derivatives. I mean, if it stopped at first derivative, it's somewhat understandable. But if a proposition proceeded to second derivative, why doesn't it just go to third, fourth, fifth... derivatives?

I'm wondering that why people tend to stop at second derivative? Is there any particular theorems that second derivative was somewhat sufficient for some special/nontrivial conditions?

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    $\begingroup$ If you consider the role of derivatives in the Taylor series it's not hard to see that terms after the 2nd derivative are very small in comparison to the first few terms. This is enough for most applications such as optics since the measurement errors tend to be larger than the remaining terms in the Taylor series. $\endgroup$ – CyclotomicField Feb 16 at 22:39

One (unsatisfying) answer is that there just aren't that many differential equations we care about with more than 2 derivatives. In the case of ODEs, higher order systems can always be reduced to a 1st order vector ODE, so many systems are only analyzed in their 1st order representations. However, there are notable exceptions (often coming from physics) where 2nd order structure is preferred, such as Newtonian systems, $$y'' = f(y).$$ A natural question then is why are physical systems often 2nd order, which has some interesting discussion here.

If you look at partial differential equations (PDEs), then there are some higher order equations that are very interesting, for example, the KdV equation $$ \frac{\partial u}{\partial t}+\frac{\partial^3 u}{\partial x^3}-6u\frac{\partial u}{\partial x}=0, $$ which appears in the study of shallow water waves, or the biharmonic equation (or Euler beam equation) $$ \frac{\partial^4 u}{\partial x^4} = 0, $$ which appears in solid mechanics when studying some thin surfaces such as plates.

It is worth noting that these higher order derivatives aren't "inherently physical," by which I mean they often come up by combining physical laws (often 2nd order) with some assumption on material behavior that then allows one to close the system and possibly end up with a system with higher derivatives. For an example, when deriving the equations of solid mechanics for a thin plate you end up with a term like $\frac{\partial^2 M}{\partial x^2}$, where $M$ is a bending moment. This relation comes from careful analysis of the system and application of Newton's Laws. One can then close the system by making some assumptions on the material properties, such as linear elasticity, to derive the relation $M\propto\frac{\partial^2 u}{\partial x^2}$. One can then combine both relations to obtain a 4th order system, even though no physical law explicitly stated a 4th order relationship between any of the quantities involved.


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