What Mathematics questions can be better solved with concepts from Physics?

Solution 1:

Personally speaking, I very rarely use Physics directly to solve Mathematics problems (apart from your example, which I used before). I do find that with very (mathematically) abstract PDE's/dynamical systems, a basic understanding of physics greatly enhances my intuition on the subject. So it helps me indirectly, but in a powerful way.

To give you a better example though, consider an analytic (read: differentiable) function $f(z)$ from $\mathbb{C}$ to $\mathbb{C}$. Many first time students struggle to visualise what this function represents, and in particular what the integral of the function represents. What does $\int\! f(z) \, \mathrm{d}z$ mean?

Physically there is a beautiful answer. Define $\overline{f(z)}$ to be the Polya vector field of $f$, where $f$ is analytic. Then in the complex plane, $\overline{f}$ is a sourceless, irrotational vector field. Astounding! What's more: $$\int_C f(z) dz=\mathcal{W}[\overline{f},C]+i\mathcal{F}[\overline{f},C]$$ where $C$ is any arbitrary curve in the plane, $\mathcal{W}$ denotes the work done along $C$ and $\mathcal{F}$ denotes the flux passing through $C$. So now complex integrals should be second nature to any physicist, particularly those who like Electromagnetism. See here for more: http://demonstrations.wolfram.com/PolyaVectorFieldsAndComplexIntegrationAlongClosedCurves/

As an example, Cauchy's Theorem states that $$\int_C f(z) dz=2\pi i \sum\limits_i Res[f(z),z_{i}]$$ where $C$ is a simple closed contour. This is a nice result, but isn't very intuitive... at least initially. Recall that $\frac{1}{z}=\frac{\bar{z}}{\lvert z \rvert ^2}$ so the Polya vector field of $f(z)=\frac{1}{z}$ is $\frac{z}{\lvert z \rvert ^2}$ which corresponds to a point charge at the origin. Now we know by Gauss' law that the flux through any closed contour should be equal to the some of the charge inside, proving Cauchy's theorem! For more on this overlap, see Tristan Needham's Visual Complex Analysis.

Solution 2:

Volume of Sphere deduced by Archimedes using mechanics concepts: Archimedes methold.

In my opinion the Archimedes method to deduce the volume of the sphere is a beatiful examples of applications of physics concepts to a mathematical proof. I will not plagiarize the method describing it here but I will indicate sites that describe the method.

I hope helped.

See the Archimedes method in this site, Youtube and pr$\infty$f wiki.

Solution 3:

The existence of a nonconstant meromorphic function on a compact Riemann surface. A priori, it is not clear that such a function necessarily exists, because one cannot "patch" meromorphic functions together with partitions of unity, like one can do on differentiable manifolds.

Riemann understood that every compact Riemann (!) surface admitted a nonconstant meromorphic function by an essentially physical intuition. The real and imaginary parts of such a function are harmonic functions away from the poles, so the question boils down to the construction of a nonconstant harmonic function. Such a function is essentially a stable energy or charge distribution: one which will not evolve or "smooth out" over time.

In order to construct such a distribution physically, one could place a number of point charges and point sinks on the surface, corresponding to poles and zeroes respectively. Imagine that the surface is a conductive mesh, and that we are making a current go through it through wires attached at different points. Provided that there are as many sinks as sources, our intuition indicates that the flow of current will stabilize in finite time, and the final distribution will be a nonconstant harmonic function, which we can take as the real part of a meromorphic function, its conjugate being the imaginary part.

The same physical interpretation gives a clear picture of Liouville's theorem, which states that every nonconstant meromorphic function on a compact Riemann surface must have a pole. Indeed, without a source, the charges will tend to even out, and after a finite time the charge distribution will be uniform, yielding a constant function.

(Of course, this is essentially the same idea as dtldarek's example of minimal surfaces. )