Determinism Without Prediction: Chaos, KAM Theory and the Fate of Mercury — Epoche C2
Whether Mercury will still be orbiting between Venus and the Sun five billion years from now is an open question, and the reason is not any gap in Newton's law of gravitation. Laplace, in the Essai philosophique sur les probabilités (1814), claimed that an intelligence knowing every body's position and velocity together with the forces acting on them could compute the entire future. The Solar System is the ideal test case for that claim: eight planets, one dominant central mass carrying about 99.86 per cent of the system's mass, a force law verified to high accuracy, no dissipation worth mentioning on the relevant timescale. The claim nonetheless fails in practice, and the way it fails is instructive, because determinism and predictability turn out to be different properties and the distance between them is measured in time. Where perturbation theory breaks: the small divisors The natural attack is perturbation theory, and it is worth setting out exactly where it fails, because everything afterwards is an attempt to work round that one point. Write the system in action-angle variables for the unperturbed problem — each planet on a fixed Keplerian ellipse about the Sun — so that the Hamiltonian takes the form $$H(I,\theta) = H_0(I) + \varepsilon H_1(I,\theta),$$ with actions $I$ in an open subset of $\mathbb{R}^n$, angles $\theta$ on the $n$-torus, and $\varepsilon$ measuring the size of the mutual planetary attractions relative to the solar one. For Jupiter that ratio is $m_J / M_\odot = 9.5\times10^{-4}$, which by the standards of asymptotic analysis is small. The unperturbed frequencies are $\omega(I) = \partial H_0/\partial I$, and the unperturbed motion is quasi-periodic: each angle advances linearly at its own rate. The classical programme is to remove the angle dependence order by order. Seek a canonical transformation generated by $S(I',\theta) = I'\cdot\theta + \varepsilon S_1(I',\theta)$ such that the new Hamiltonian depends on the new actions alone to first order. Matching terms gives the homological equation $$\omega(I')\cdot \frac{\partial S_1}{\partial \theta} + H_1(I',\theta) = \langle H_1\rangle(I'),$$ where the angular bracket denotes the average of $H_1$ over the angles. Expanding the perturbation in its Fourier series, $H_1 = \sum_{k\in\mathbb{Z}^n} h_k(I')\,e^{ik\cdot\theta}$, the equation is solved term by term: $$S_1(I',\theta) = \sum_{k\ne 0} \frac{i\,h_k(I')}{k\cdot\omega(I')}\,e^{ik\cdot\theta}.$$ There is the whole difficulty, in one line. Every Fourier mode acquires the denominator $k\cdot\omega$, an integer combination of the orbital frequencies. The same denominators appear in the older and more elementary route of integrating a term $\cos(k\cdot\theta)$ along the unperturbed motion, which returns $\sin(k\cdot\omega t)/(k\cdot\omega)$; the generating-function version simply makes clear that the denominators are not an artefact of a particular bookkeeping. Whenever some integer combination of the frequencies is near zero — a resonance — the corresponding "correction" is enormous, and since the vectors $k$ range over all of $\mathbb{Z}^n$, frequencies for which $k\cdot\omega$ vanishes exactly are dense in the frequency space. Density alone would not be fatal if the numerators fell fast enough. For a real-analytic perturbation the coefficients decay geometrically, $|h_k| \lesssim e^{-\sigma|k|}$ for some $\sigma > 0$, so the question is whether $e^{-\sigma|k|}/|k\cdot\omega|$ remains summable over $k$. That is a competition between exponential decay in the numerator and an unbounded, erratically fluctuating denominator, and it is not decided by the size of $\varepsilon$. Poincaré, in Les Méthodes Nouvelles de la Mécanique Céleste (1892–1899), settled the classical form of the question: the series produced by this scheme do not in general converge, and the three-body problem admits no complete set of analytic first integrals independent of energy and angular momentum, so the older hope of solving it by finding enough conserved quantities is not merely hard but unavailable. KAM theory: excluding the dangerous frequencies rather than the resonant terms The Kolmogorov–Arnold–Moser theorem rescues part of the construction by a change of strategy: instead of trying to make the series converge for all initial conditions, it discards the initial conditions whose frequencies are badly behaved and proves convergence for the rest. A frequency vector $\omega$ is called Diophantine with constants $\gamma > 0$ and $\tau$ if $$|k\cdot\omega| \ \ge\ \gamma\,|k|^{-\tau} \qquad \text{for every } k \in \mathbb{Z}^n\setminus\{0\},$$ which says that no integer combination of the frequencies comes too close to zero, with the permitted closeness shrinking only polynomially in $|k|$. This is exactly the condition that restores summability: the general term is then bounded by $\gamma^{-1}|k|^{\tau}e^{-\sigma|k|}$, and a polynomial cannot defeat a geometric decay. The condition also explains why $\tau$ must exceed $n-1$, which is worth doing explicitly because the exponent is otherwise mysterious. For a fixed $k$, the set of $\omega$ in a bounded domain violating the inequality is a slab about the hyperplane $k\cdot\omega = 0$ of thickness $2\gamma|k|^{-\tau}/|k|$, hence of measure of order $\gamma|k|^{-\tau-1}$. There are of order $R^{n-1}$ integer vectors with $|k|$ near $R$, so summing over all $k$ gives a total of order $\gamma\sum_R R^{n-1}R^{-\tau-1} = \gamma\sum_R R^{n-2-\tau}$, which converges precisely when $\tau > n-1$. With $\tau$ so chosen, the measure of the excluded set is of order $\gamma$ and can be made as small as one likes by taking $\gamma$ small — at the cost, as will matter shortly, of weakening the theorem's smallness threshold. The theorem itself, in the form Kolmogorov announced in 1954 and Arnold and Moser proved, requires three hypotheses and it is worth naming all of them, because dropping any one changes the conclusion. The Hamiltonian must be sufficiently smo