by S.Yu. Orevkov
We give a physical (in fact, a variational) proof of the following Whitney's theorem:
Theorem. Two immersions of a circle to the plane are regularly homotopic if and only if they have the same winding number.
The winding number of an immersion of a circle to the plane is the increment of the argument (i.e. the number of twists) of the velocity vector.
To prove this theorem, we find more or less explicitely all extremals of the energy functional on the space of immersed curves of a fixed length. The energy of an immersed curve is defined as the integral of the square of the curvature. Solving Euler-Lagrange equation, we show that all critical points of the energy are circle and the "figure eight" (possibly, passed several times).
Physically, this functional can be interpreted as the potential energy of
an elastic (steel) wire which is placed in a gap between two plates.