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As a first approximation, neglect lower order terms

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Quick description

Often in analysis, one is faced with a complicated expression (e.g. an integral or a partial differential equation) involving many different terms, and it may not be immediately clear how to proceed to simplify or otherwise control the expression. However, in many cases one can assign different terms a different "order" which measures how large or how troublesome the term is expected to be. Then one way to proceed is to pretend that one can automatically ignore all but the "highest order" or "worst" terms, and solve that simplified problem first. After the highest order terms are controlled adequately, one can then insert back in the next highest order terms into the argument, and control them also; and continue in this fashion until one returns to the original problem. This way, one only deals with one level of difficulty at a time, rather than all of them at once. (Also, there is usually very little point working very hard on the lower order terms unless the higher order terms are already under control.)

The precise meaning of "higher order" and "lower order" terms varies from context to context. But here are some examples:

  • If there is an important large parameter N in play, then terms which are quadratic in N (e.g. aN^2) could be considered higher order than those that are linear in N, terms which are cubic in N would be considered higher order still, and so forth. Terms which are exponential in N (e.g. e^{cN}) would be higher order than all polynomial terms, while terms which are logarithmic in N (e.g. \log^k N) would be lower than all polynomial terms, and so forth.

  • Similarly, if there is an important small parameter \varepsilon in play, terms that are independent of \varepsilon would be "higher order" than those that are linear in \varepsilon, which in turn would be higher order than those that are quadratic in \varepsilon, etc. (In microlocal analysis and quantum mechanics, the symbol h or \hbar is often used for the small parameter.) Note that in this case, the smaller powers of \varepsilon are often referred to as "lower order" terms rather than "higher order" terms.

  • If there is a low regularity function u in play, then terms involving first derivatives \nabla u of u can be considered higher order than those involving zeroth derivatives u of u, but are in turn lower order than terms involving second derivatives \nabla^2 u of u. This type of situation is particularly common in nonlinear PDE.

  • If there is a nilpotent group G in play, elements in G can be considered higher order than elements in the first commutator group G_2 = [G,G], which in turn are higher order than elements in the second commutator group G_3 = [G,G_2], etc. (See also "First pretend that a normal subgroup is trivial")

Prerequisites

Analysis; partial differential equations; group theory

Example 1

(Energy estimates exploiting cancellations for some nonlinear PDE)

(Some example of a computation which is easier to follow if one first "ignores log factors")

Suggestions welcome!

General discussion

Comments

First order approximations and neglecting higher order terms

An example of neglecting higher order terms in a computation to simplify it is given in Tips on Physics. Page 77 of this book lists a computation by Feynman of the deflection angle of a fast moving charged particle in a central field generated by a fixed electrical charge. The calculation is short, simple and beautiful and gives the right answer up to a constant factor. His approximation ignores the interactions of the particles when they are far away, relativity and the horizontal interaction when the particles are near. He explains why it is a good idea to ignore these things. A precise computation requires at least the study of a Hamilton-Jacobi equation.

As is, it seems as if this example doesn't quite fit this page because it is not exactly an example of the method described here. But it is clearly related to the subject of this page.

Here are some further thoughts/sentences on ignoring higher order terms:
We are often confronted with problems, which may have very complicated models. When this is the case, don't go first to the most complicated model but to a simpler one that approximates the real model at least when certain parameters are close to some limit values. Do your computations first for this simple model. The solution of the simpler model can be helpful in several ways: 1) It can be an approximation to the initial model. One can then get some idea how the original solution behaves by looking at the behavior of this approximate solution. This is useful, because it gives us an idea about what we should look for in a real solution 2) Sometimes, we get super lucky and the approximate solution tells us the exact nature of the functional dependence of the real solution to some of the important parameters of the system. This seems to be the case, for example, for the deflection angle problem.

Any ideas on how and whether to incorporate these thoughts and the deflection angle example into this article or into tricki in general?

These also seem to be related to other subjects in tricki, including convergence and approximation. Perhaps, we can think about how to organize/link articles on these subjects.