This is a fascinating
question that really illustrates a lot about the structure of nature.
TL;DR: The answer is that
higher order differential questions appear ubiquitously in physics, but rarely
in an important way.
Ubiquity of Higher Order
Derivatives
So first, derivatives of
all orders appear in Taylor series. So anytime you’re dealing with
approximations, you’re likely going to run into Taylor series or approximation
methods that are morally equivalent to a Taylor series. So therefore, you’re
going to run into higher order derivatives all over the place. This includes
both classical and quantum mechanical perturbative methods.
Almost nothing can be
solved exactly in physics — except for the the harmonic oscillator. This means
that you’re going to run into higher order derivatives all the time if you go
to high enough order in the approximation. These higher order termes can be
essential for getting the right numerical answer.
I would call these higher
derivatives inessential. Sure, you need to know about higher order derivatives
(which aren’t anything special), but they don’t change anything conceptual
aspects of the problem.
Important Higher Order
Derivatives
Essential higher order
derivatives are when you have to solve a higher order differential equation.
These are few and far between in physics, but they do occur. The biharmonic
equation:
∇2∇2ϕ=0∇2∇2ϕ=0
is one equation that
occasionally appears in physics, most notably in continuum mechanics. You may
have to solve a fourth order differential equation.
The reason these higher
order differential equations are few and far between is because generally there
is nothing that forbids the lower order differential operator from appearing in
the equation, so generically you’d expect
(a∇2+b∇4)ϕ=0(a∇2+b∇4)ϕ=0
So
this is a fourth order differential equation still, but notice that there are
units associated with the constants aa and bb , in particular
dim (b/a)=distance2dim (b/a)=distance2
In most systems, this
ratio with be some short distance scale having to do with atomic size (or some
other short distance phenomena); however, the equations you’re solving are only
expectation for the lowest order behavior
of most physical systems.valid for much longer distances. So the
higher order derivative is going to make contributions of the size
δ∼(b/al2)≪1δ∼(b/al2)≪1
relative
to the first term where ll is the size of the phenomena that is being described.
If you insist on taking
that higher order term seriously and start solving a fourth order differential
equation, you have to explain why the next higher derivative doesn’t matter
(a∇2+b∇4+c∇6+d∇8+⋯)ϕ=0(a∇2+b∇4+c∇6+d∇8+⋯)ϕ=0
where the relative sizes
of these different operators is generally
b/a∼c/b∼d/cb/a∼c/b∼d/c
so when the fourth order
operator starts becoming important, all the different orders start becoming
important at the same time and the equation isn't a good approximation to the
phenomena being described.
In some systems you can
either have accidentally small or large coefficients — or in some modern
condensed matter or optical systems, you can tune parameters to be large or
small. In these cases you can choose to end up with higher order differential
equations. Some of these are being constructed to study novel physical
phenomena where the ordinary harmonic oscillator isn’t the lowest order behavior.
But
aside from these unusual situations, the near-guarantee of the ∇2∇2 term dominating the long-distance behavior of physical systems is the
reason when higher order derivatives are rarely discussed or used, they can
usually only be treated as perturbations to the original equation and not as
the core parts of the equation.
There are these rare
exceptions, oftentimes due to weird symmetries, where higher order differential
equations appear, but these are the exception not the rule.
And ultimately this is
why the harmonic oscillator is the most important model in physics: because it
forms the basis for all perturbative methods because it is the generic
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