Good examples of Ansätze

Question

Frequently, when talking to mathematicians, I have some trouble when I mention, use, or try to explain what an Ansatz is. (Apparently it is more of a physics term than a maths one, for some reason.) The Wikipedia page on it has what I think is a good definition:

an educated guess that is verified later by its results.

It also has one example which really resonates with the way I normally see the term being used, namely exponential Ansätze for the solutions of a differential equation. There one has a problem such as $$y''(x)+ay'(x)+by(x)=0,$$ and one quite nonchalantly assumes that the solution is $y(x)=e^{kx}$, leaving some leeway into not specifying $k$. This (Ansatz) is of course unjustified, and the only rigorous explanation for what one is doing is that one is blindly testing to see if a function of that form can be one particular solution.

One then, of course, goes on to show that this is indeed de case when $k$ satisfies $k^2+ak+b=0$, and this usually yields two distinct roots $k_1$ and $k_2$ with associated linearly independent solutions. The upshot of this is that one can now something very general about any solution of the original problem - i.e. that it is of the form $$y=A e^{k_1 x}+B e^{k_2x}$$ for unspecified complex coefficients $A$ and $B$ - from the original, very limited Ansatz.

---

While this example is nice, I can't think of other simple, strong examples of this sort of argument, where a simple and limited educated guess turns out, at the end, to encapsulate the whole generality of the problem; I would like to see more of those.

Answer 1

Perhaps this example is too simple. Think about the wave equation, let's say on the real line: $$ \frac {\partial^2 u}{\partial t^2} = c^2 \frac {\partial^2 u}{\partial x^2}.$$ Let's imagine that we have derived this as the equation governing wave propagation with propagation speed $c$. Physically one expects the simplest solutions to be travelling waves. If $u$ is a wave travelling right (resp left) with speed $c$, then $u$ is a function only of $x-ct$ (resp $x+ct$). So it is meaningful to make the Ansätze $u = f(x-ct)$ and $u = g(x+ct)$ for some $C^2$ functions $f$ and $g$, and once one has the Ansatz it's easy to see that we have now completely solved the equation.

Answer 2

The method that I most strongly associate with the word Ansatz (full disclosure: I'm German) is the Ritz method. In fact the Wikipedia article uses the term "Ritz ansatz function" for what is also known as a "trial wavefunction". This shows that the Wikipedia definition as "an educated guess that is verified later by its results" doesn't cover the entire concept; in this case, the wavefunction is known to be more complicated than the ansatz, and the ansatz is made not in order to be verified but to get as close as possible to the exact result.

Answer 3

I am not a great fan of the notion of ansatz, simply because it seems magical.

One can but imagine that most ansätze are the result of whole series of unsuccessful tries, and that we only hear of the successful ones. So whenever I read «let us try the ansatz such and such» I understand «ok, I spent a couple of weeks trying stuff that did not go anywhere but, who knows how, finally managed to make it work, so let me just tell you the short story and, by the by, make myself look like I came up with this stuff out of the blue»

Answer 4

Separation of variables for linear partial differential equations. The ansatz is the guess that a solution can be written as a tensor product of functions that depend separately on individual coordinate values.

Answer 5

Solving the difference equation $$x_{n+1} = ax_n + b$$ a natural guess since the numbers are involving repeated multiplication is $a^n$. But this forces $b=0$, so we modify our ansatz with a constant $$ x_n = a^n + c. $$ This gives $$ a^{n+1} + c = a^{n+1} + ac + b,$$ so $c = b/(1-a)$.

More generally, I think they are most common when solving equations, and is used to reduce degrees of freedom! The guesses are most certainly not random, even though their motivation might have been lost.

Answer 6

How about the integral $$ \int \sec x dx = \int \sec x \frac{\tan x + \sec x}{\tan x + \sec x}dx, $$ with the next step being the u-substitution $u = \tan x + \sec x$?