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

Over the years, I've seen several questions in mathematics that can be solved using concepts borrowed from Physics. Having seen these question, I'm interested to find out what other mathematics questions you've found that can be better solved with a concept from physics - or at least where the application of physics is interesting and perhaps illuminating.

Examples
One of these questions is on minimizing the time taken for a lifeguard to go out to a stationary distressed swimmer. In the scenario, the lifeguard runs faster than he swims in the water, and as such the straight line is not the fastest way for the lifeguard to reach the swimmer. Students will normally use calculus to solve this problem, and the answer can be obtained after some work - however, a much more convenient (and intuitive) way is to borrow from the idea of refractive indices in geometric optics. We recast the situation by replacing the beach and the sea with two materials with different refractive indices, choosing the appropriate refractive indices proportionate to the ratio between the lifeguard's velocities while running and swimming. The problem is then reduced to finding a beam of light that passes through both the swimmer and the lifeguard's position. (For a more complete explanation, you can visit this site: http://findingmoonshine.blogspot.sg/2012_05_01_archive.html)

Another of these questions requires one to prove that, in an acute-angled triangle, the angle subtended by any side of the triangle at the Toricelli point is 120°. Again, instead of using trigonometry, one can use the concept of hanging equal weights from a (frictionless) string at each of the vertices of the triangle, and then tying each of the three strings together at one knot placed on the surface of the triangle. The equilibrium position of the knot is the Toricelli point, and one can then complete the proof by considering forces acting on the knot.

Looking forward to hearing from you!

• This post asking for the connections between math and physics might also be of interest. Dec 10, 2012 at 14:39
• Just noting, your example is pretty "circular" because to prove the property of refractive indices you do the calculus you mention. So its not really a valid example unfortunately. Dec 10, 2012 at 17:46
• @dinoboy are you referring to the principle of least time? Dec 11, 2012 at 11:28
• No, the principle of least time requires no time. However, the route derived using the principle of least time requires calculus to prove it is indeed the route of least time I believe. Dec 11, 2012 at 17:33
• I can't think of a good answer, but there's gotta be something in lie algebras. and probably something using Euler angles. Nov 13, 2013 at 3:16

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.

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.

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. )

• Bruno - I'm not sure I have the mathematical ability to explain your answer fully if asked about it, but thank you for spending the time to write the answer so long after the question was first asked! Nov 9, 2013 at 5:24
• Dear @VincentTjeng My pleasure. The question showed up on the front page, so I figured I would contribute. Regards, Nov 12, 2013 at 17:53

Minimal surfaces using, for example, soap bubbles.

Maximum flow using actual pipes.

Optimal space filling and tesselations inspired by honeycombs.

Solving some of equation sets can be done via circuits of resistors.

There were analog computers build to analyze differential equations.

Probability can be calculated by experiments, dice games were examples of this.

Moreover, a lot of regular operations like addition, multiplication, squares and roots, powers, integration, etc., can be done by experiments.

Finally, modern computers use physics (i.e. are real) and can solve a variety of math problems :-)

Cheers!

The most interesting such result that I have seen is the connection between the moments of the Riemann zeta function and quantum energy shells.

The story goes that Brian Conrey challenged quantum physicists to use a proposed relationship between the Zeta function and quantum mechanics to continue a sequence relating to the moments of the zeta function (using solely mathematical methods, an explicit formula had not (at the time) been found). Jon Keating, a physicist, continued the sequence; in fact, he developed a formula for the terms of said sequence.

• That's really interesting! I can't think for the life of me why primes and quantum energy should be related, though :) Nov 9, 2013 at 5:26

The work done by Donaldson on 4-manifolds used techniques borrowed from gauge theory. Witten also used techniques from Physics to solve mathematical problem, including if I remember correctly, a simpler proof of the index theorem using supersymmetry.

The solution of the Poincaré Conjecture in topology by Perelman involved the introduction of a suitable entropy functional which behaves monotonically under the Ricci flow. Entropy is a concept coming from physics, statistical mechanics to be more specific.

Somewhat related example: the Kolmogorov-Sinai entropy distinguishes dynamical systems given by Bernoulli shifts (Ornstein's Theorem).

A problem in geometry. Let $$P$$ be a point inside a triangle $$ABC$$, and $$P_a, P_b, P_c$$ its projections onto $$BC, CA, AB$$ correspondingly. Prove that at least one of the projections lies on the side itself, i.e. $$P_a \in BC$$ or $$P_b \in CA$$ or $$P_c \in AB$$.

This is a very easy (almost trivial) problem, but it has a beatiful solution using concepts from Physics.

Solution: By contradiction, let's put some weight weight into $$P$$ and put the triangle onto one of its sides. Since the center of mass is in $$P$$, the triangle will flip onto other side under its mass. This process is going to be perpetual, which means we constructed perpetuum mobile, a contradiction