Energy functionals for knots and plane curves, and their normal forms

Let us supply the moduli space of all $\mathcal C^2$ curves in $\mathbb R^2$ with the Euler functional
$$ E(\gamma) = \int _0^{2\pi}\, \big( \kappa(\gamma(s)) \big)^2 ds, $$
where $\gamma\colon\mathbb S^1 \to \mathbb R^2$ is a curve of length $2\pi$, $s$ is the arclength parameter, and $\kappa(\gamma(s))$ is the curvature of $\gamma$ at the point $s$; we study the critical points (curves) and the local minima of this functional, and call the curves corresponding to local minima normal forms.
Theorem 1. (i) A critical curve of the Euler functional is either a circle passed once or several times, or Bernoulli's lemniscate $(\infty$-shaped curve$)$ passed once or several times.
(ii) A Bernoulli lemniscate passed more than once is not stable.
(iii) A circle passed once or several times and a Bernoulli lemniscate passed once are local minima.
This theorem gives a solution of the so-called Euler problem for plane curves, set in 1754. The same result has recently been obtained by Yu. Sachkov, but by a more laborious method. Our proof uses the Gauss representation of planar curves, classical methods of the calculus of variations, elliptic integrals, and some more recent ideas from functional analysis, e.g. the Dirac $\delta$-function. We also prove the following theorem.
Theorem 2. Two regular plane curves of class $\mathcal C^2$ are regularly homotopic if and only if they have the same normal form with respect to the Euler functional $E$.
This theorem is the “mechanical form” of the classical Whitney–Graustein theorem.
Our approach can be carried over to three-dimensional knots: the functional that we use in that case is $E+R$, the sum of the Euler functional and a repulsive functional $R$, which prevents self tangencies and crossing changes. A discretized version of gradient descent along the functional $E+R$ was used to design a computer program, implemented in an animation that shows how curves are homotoped to the normal forms indicated in Theorem 1, and how knots are isotoped to their normal forms. We conjecture that our computer program solves the unknotting problem.
For the most part, his talk is the result of joint work with S. Avvakumov and O. Karpenkov (see [2], [3], [4]).