Mark Levi's The Mathematical Mechanic is a book of examples of how physical reasoning can be used to solve mathematical problems; another couple of examples is in this blog post at Concrete Nonsense. Many of these proofs rely on conservation laws and other physical facts which, at least as I understand it, can be made rigorous by appealing to Noether's theorem. So these proofs themselves ought to be presentable in a mathematically rigorous way. But how?
There are many examples, so I will focus my question on two, one of which is in Levi's book and one of which I recently encountered on a friend's homework.
Problem: prove the Pythagorean theorem. Solution: imagine a fish tank in the shape of a triangular prism. The triangle is a right triangle with side lengths $a, b, c$; call the vertices of the two copies of the triangle $A, B, C$ and $A', B', C'$, where the right angle is at $C$ (resp. $C'$). Drive a rod through $BB'$ and fill the tank with water. This system is at equilibrium (by conservation of... angular momentum?), so the net torque through the rod is zero. On the other hand the net torque through the rod is proportional to $c^2 - a^2 - b^2$.
Problem: on each face of a tetrahedron, not necessarily regular, place a vector pointing normal to and out from the face with magnitude equal to the area of the face. Prove that the sum of these vectors is zero. Solution: fill the tetrahedron with an ideal gas. This system is at equilibrium (by conservation of... energy?), so the net force exerted by the gas on the tetrahedron is zero. On the other hand the net force exerted on each face is proportional to its area and points normal to and out from each face.
As you can see, I'm not even totally clear what conservation law(s) I'm invoking at the crucial step of the physical arguments. I would really appreciate mathematical insight into what's going on here.