I'm looking for cool, competition-style problems that borrow ideas from physics in their solution. For example, one I found in the book Putnam and Beyond goes like this:
Orthogonal to each face of a polyhedron $S$ construct an outward vector with length numerically equal to the area of the face. Prove that the sum of all these vectors is equal to zero.
Solution:
Assume that the interior $V$ of polyhedron $S$ is filled with gas at a (not necessarily constant) pressure $p$. The force that the gas exerts on S is $\int\int_S p \:\hat{n} \: dA$, where $\hat{n}$ is the outward normal vector to the surface of the polyhedron and $dA$ is the area element. The divergence theorem implies that $$\int \int_S p\:\hat{n}\: dA = \int\int\int_V \nabla p \:dV$$ Here $\nabla p$ denotes the gradient of $p$. If the pressure p is constant, then the right-hand side is equal to zero. This is the case with our polyhedron, where $p = 1$. The double integral is exactly the sum of the vectors under discussion, these vectors being the forces exerted by pressure on the faces.
I realize that, in retrospect, it is possible to purge any physics from this solution by just jumping straight to the divergence theorem, but regardless I like the fact that you can use a physical analogy (interpreting the vectors on the faces of the polyhedron as forces) to intuit a solution. What are some other problems of this sort?
Comment 1: I asked long ago whether a request for questions was on-topic, and got a tentative yes.
Comment 2: There are a few concepts from physics which I am confident can be used to good effect for pure math problems. I'll mention a few in case it jogs someone's memory and helps them recall such a problem:
- The use of entropy/energy as a monotonic quantity in differential equations.
- Pauli matrices for rotations
- The concept of thermodynamic equilibrium (as used by Einstein to derive spontaneous emission coefficients).
- Fermat's principle for finding solutions to variational problems (as in the Brachistochrone problem).
Of course I also welcome problems that do not use these ideas!
