October 9, 2017 — Support my next blog post, buy me a coffee ☕.
My second paper as a Ph.D. student, in collaboration with my friend and former colleague at Oxford Nikola Gushterov, was about the choreographies of the \(n\)-body problem. These are very simple periodic solutions of the \(n\)-body problem, in which the bodies share a common orbit and are uniformly spread along it; see, e.g., these animations. The trivial ones are circles, and these were found by Lagrange in 1772. Fore more than two centuries, this was the full story.
Everything changed at the end of the 20th century, when Moore (numerically in 1993), and Chenciner and Montgomery (theoretically in 2000), discovered the first non-trivial choreography: the figure-eight of the three-body problem. And then the magic happened: in the early 2000s, Simò found many new choreographies using numerical optimization of the so-called action. In this post, I explain the idea of Simò, who coined the term choreographies, the \(n\) bodies being "seen to dance in a somewhat complicated way."
Let \(z_j(t)\in\mathbb{C}\), \(0\leq j\leq n-1\), denote the positions of \(n\) bodies with unit mass in the complex plane. The \(n\)-body problem describes the motion of these bodies under the action of Newton's law of gravitation, through the nonlinear coupled system of ODEs $$ z_j^{''}(t) - \sum_{\underset{i\neq j}{i=0}}^{n-1} \frac{z_i(t) - z_j(t)}{\big\vert z_i(t) - z_j(t) \big\vert^3} = 0, \;\, 0\leq j\leq n-1. $$ Since choreographies share a single orbit and are uniformly spread along it, these can be written as $$ z_j(t) = q\Big(t + \frac{2\pi j}{n}\Big), \;\, 0\leq j\leq n-1, $$ for some \(2\pi\)-periodic function \(q:[0,2\pi]\rightarrow\mathbb{C}\).
It has been well known since Poincaré that the principle of least action, first introduced by Maupertuis in 1744, can be used to characterize choreographies: these are minima of the action, defined as the integral over one period of the kinetic minus the potential energy, $$ A = \int_0^{2\pi} \big(K(t) - U(t)\big)\,dt, $$ with kinetic energy $$ K(t) = \frac{1}{2}\sum_{j=0}^{n-1} \Big\vert q'\Big(t + \frac{2\pi j}{n}\Big) \Big\vert^2 $$ and potential energy $$ U(t) = -\sum_{j=0}^{n-1}\sum_{i=0}^{j-1} \Big\vert q\Big(t + \frac{2\pi i}{n}\Big) - q\Big(t + \frac{2\pi j}{n}\Big) \Big\vert^{-1}. $$ Note that the action \(A\) depends on \(q(t)\) via \(U(t)\) and on \(q'(t)\) via \(K(t)\).
Since the integral of \(K(t)\) does not depend on \(j\) and the integral of \(U(t)\) only depends on \(i-j\), the action can be rewritten $$ \begin{align} A = \frac{n}{2}\int_0^{2\pi} \big\vert q'(t) \big\vert^2 dt + \frac{n}{2}\sum_{j=1}^{n-1} \int_0^{2\pi} \Big\vert q(t) - q\Big(t + \frac{2\pi j}{n}\Big) \Big\vert^{-1}dt. \end{align} $$ Choreographies correspond to functions \(q(t)\) which minimize \(A\). Since these are closed curves in the complex plane, these can be represented by Fourier series. The action \(A\) becomes a function of Fourier coefficients, can be computed with the exponentially accurate trapezoidal rule, and an optimization algorithm (over a finite number of Fourier coefficients) can be used to found the minima.
In our paper, we have recomputed the choreographies found by Simò to higher accuracy, and extended his work to spaces of constant positive curvature—check it out!
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