Ho fiHi iH 1111

Since the problem is integrable completely when i = 0, we can choose variables pa and qa so that H0 = H0(pa) (in which case pa = 0 ^ pa = const, and qa = dH0/dpa ^ qa = const). One such choice is the Delaunay variables q = (l, g, h), p = (L, G, H), where L = +fa, G = LV 1 — e2, H = G cos i, l is the mean anomaly, h is the longitude of ascending node, g + h is the longitude of perihelion, and then each of the terms Hi is periodic in the variables l, g, h. The study of Hamiltonian systems in which the Hamiltonian is of the form of Eqn (11.11), with H0 = H0(pa )and Hi periodic in the variables qa ,is called the 'general problem of dynamics' by Poincare. Nowadays, the variables pa are called 'action variables', and qa are 'angle variables'.

When expanding the perturbing function, Poincare found that it was often convenient to work in terms of a different set of variables obtained from the Delaunay variables through two canonical transformations. First, we replace the angle variables l, g, h by l + g + h, -g - h, -h, and then the action variables L, G, H have to be replaced by L, L - G, G - H so as to preserve the Hamiltonian nature of the system. Then we can apply the transformation defined by Eqn (11.5) to the second and third conjugate pairs, to obtain what are now referred to as 'Poincare variables':

This new set of canonical elements turns out to be particularly advantageous when e and i are small.

Poincare devoted considerable effort toward finding periodic solutions to the restricted three-body problem. The first such solutions to be found were the straight line and equilateral triangle solutions of Euler and Lagrange, respectively. Hill's intermediate orbit for the lunar theory was another example, which Poincare generalized in a paper in 1884 when he showed that a whole family of periodic solutions existed for the case when two of the masses were very small. These solutions could be classified into three different categories and, in New Methods, Poincare described their relation to two-body solutions. First-type solutions, in which the eccentricity is small and the inclination is zero, are continuations of circular solutions to the two-body problem to the case i > 0, elliptic two-body orbits generate periodic solutions of the second type, and third-type solutions come from elliptic two-body solutions when the small mass i is not in the orbital plane. Hill's periodic solution is an example of the first type. In all three cases, the mutual distances between the bodies are periodic functions of time, so that at the end of a period the bodies are in the same position relative to each other. However, the system will have turned through a certain angle. In order for the coordinates of the bodies to be periodic functions of time, we would need to refer them to a set of uniformly rotating axes.

Periodic orbits are special because we can know everything about them by studying only a finite amount of time. But is knowledge of them useful? In general, the probability of a given initial state leading to a periodic orbit is zero, but Poincare realized that they might usefully be used as intermediate orbits in the style of Hill, and held out the hope that one can always find a periodic solution that approximates any given solution to a given accuracy over as long a time as desired. He described periodic solutions as 'the only breach by which we may attempt to enter an area heretofore deemed inaccessible'.39

There are conditions under which periodic solutions do not persist when / is increased from zero, and this can also be used to advantage. Suppose we have three bodies of mass 1 - /, and zero, and suppose in the first instance that / = 0, with the two massless bodies orbiting the third in circles with mean motions n and n' (n' > n). This system is periodic with respect to axes rotating with angular speed n, with period 2n/(n' - n). Poincare showed that periodic first-type solutions will exist when / is small but non-zero, provided n'/(n' - n)

is not equal to an integer. In other words, we cannot have the case

n' j for an integer j. However, what happens if n'/(n' - n) is not integral, but only nearly so? In this case, there will be a first-type solution, but it will exhibit a large irregularity. In the case, of the Saturnian moons Hyperion and Titan, their mean motions are roughly in the ratio 3 : 4 (nH/nT = 0.749 43 ...) and we would expect this to lead to a considerable inequality in their motion. Whatever the actual motion of the two satellites, the form of this inequality will be established

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