Since the beginning of MPE2013, I have met people who were surprised when I classified celestial mechanics as a topic that would fit under Mathematics of Planet Earth. But part of celestial mechanics is concerned with planetary motion, and Earth is a planet.
The toy model for planetary motion is the n-body problem, which describes the motion of n massive particles subject to Newton’s gravitational law. The n-body problem is a purely mathematical problem. The model consists of a system of 3n second-order differential equations. Using the Hamiltonian formalism, one transforms the system into a system of ordinary differential equations in dimension 6n.
In the 19th century, mathematicians were looking for first integrals, i.e., quantities that would remain constant along the trajectories of the system. Because we expect quasi-periodic solutions (i.e., superpositions of periodic solutions of different periods for the different planets), Weierstrass computed solutions in the form of Fourier series, but he could not show their convergence. The field of the n-body problem experienced its first revolution in 1885, when Poincaré showed that the system was not integrable and that there exist chaotic solutions for which the Weierstrass’ series diverge. The second revolution came with KAM theory in the 1950s and ’60s, following earlier work by Carl Siegel. KAM stands for Kolmogorov, Arnold and Moser. Kolmogorov essentially conjectured the results in 1954, which were later proved in the ’60s by Arnold in the analytic case and by Moser in the smooth case.
KAM theory is concerned with systems that are close to an integrable system. The solar system is integrable if we neglect the mutual interactions among the planets. Then each planet has a periodic orbit around the Sun, and the system as a whole is quasi-periodic. Since the interaction between planets is small, we are relatively close to an integrable system. What happens then? It depends on the periods of the planets. If the periods of the planets are commensurable, the systems comes back regularly to the same position and the perturbations add up. We call this the resonant case; the corresponding set of initial conditions has measure zero. If we are sufficiently close to the resonant case we are again in the chaotic regime, and if we are far from the resonant case we expect stability. Hence, the initial conditions are intertwined with an open dense set of resonant initial conditions of very small measure corresponding to chaotic motions and a set of nearly full measure for which we have stable quasi-periodic motions. To the famous question: “Is the solar system stable?” we would have answered in the 1970s: “Yes, if we have the right initial condition.”
Now we know more. We know that the solar system is too far from an integrable system to enable us to apply KAM theory directly. But the spirit of KAM theory remains. We know that resonances are responsible for chaotic motions, and when we find some chaotic motions we look for the resonances that could have been responsible for them.
Jacques Laskar made an extensive study of the solar system. In 1994 he gave numerical evidence that the inner planets (Mercury, Venus, Earth and Mars) have chaotic motions and identified the resonances responsible for their chaotic behavior. Because of the sensitivity to initial conditions, numerical errors grow exponentially, so it is impossible to control the positions of the planets over long periods of time (hundreds of millions of years). In his simulations, Laskar used therefore an averaged system of equations. The simulations showed that the orbit of Mercury could cross that of Venus for some period of time. Laskar could explain this chaotic behavior by exhibiting resonances in some periodic motions of the orbits of the inner planets.
Another way to study chaotic systems is to make numerous simulations in parallel with close initial conditions and deriving probabilities of future behaviors. The shadowing lemma guarantees that a simulated trajectory resembles a real trajectory for a close initial condition. In a letter published in Nature ["Existence of collisional trajectories of Mercury, Mars and Venus with the Earth," Nature 459, 817-819 (11 June 2009) | doi:10.1038/nature080962009], Laskar and M. Gastineau announced the results of an ambitious program of 2000 parallel simulations of the solar system over periods of the order of 5 billion years. The new model of the solar system was much more sophisticated and included some relativistic effects. The simulations showed a 1% chance that Mercury could be destabilized and encounter a collision with the Sun or Venus. A much smaller number of simulations showed that all the inner planets could be destabilized, with a potential collision between the Earth and either Venus or Mars, in around 3.3 billion years.