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force, and its meaning was the subject of much debate. Equations (9.12) are purely kinematic in character, arrived at by formal manipulations of various derivatives, but now we can incorporate the dynamics by utilizing the equations of motion.

Lagrange introduced what we would now call a 'potential function'. For any system of n mutually interacting particles, it is possible to determine a function, V, of the positions of the masses (the 'potential energy' as we would now call

it, but called the 'perturbing function' by Laplace) such that the equations of motion can be written (using modern notation for brevity) in the form m, r, = _V,V, i = \,...,n, (9.13)

in which V, = + j -¡y + k. For the three-body problem in Eqn (9.2),

Incorporating Eqn (9.13) into Eqn (9.12) yields d d T d T ^ d ri d V ----= _VvV • — =--, a = 1,..., N.

dt d qa dqa d qa dqa

It is customary now to define a new function L (qa, qa) - the Lagrangian - by

dt d qa dqa

These are Lagrange's equations of motion, and bear all the hallmarks of Lagrange's mathematical style through their elegance of form and generality of application. They reduce the derivation of the equations of motion to the construction of a single scalar function, L, and could be applied to the «-body problem (in which case N = 3«) whatever coordinates qa with which one chose to work. In his masterpiece Le mecanique analytique (Analytical Mechanics) (1788), Lagrange illustrated the use of these equations with a number of applications and wrote:

These different examples comprise nearly all the problems on the motion of a body or a system of bodies that the Geometers have solved; we have chosen them on purpose so that one may better judge the advantages of our method, by comparing our solutions with those found in the works of Messr. Euler, Clairaut, d'Alembert etc., in which one arrives at the differential equations only by reasonings, constructions and analyses often rather long and complicated. The uniformity and the swiftness of the course of [our] method are what should principally distinguish it from all others, and what we wished especially to show in these applications.66

By an expeditious choice of variables, Lagrange derived differential equations that enabled him to show that the angular distance of the Moon from the mean ascending node of its orbit had to be the same as the mean angular distance of the prime meridian of the Moon from the node of the lunar equator. This is

65 First done by Poisson in 1809 (Dugas (1988), p. 384).

Lagrange Analytical Mechanics, Section 5, Part 2. Quoted from Fraser (1983).

precisely what Cassini had observed, and thus once more, what had presented itself as an apparently arbitrary fact turned out to be a consequence of universal gravitation.

### Exact solutions to the three-body problem

As well as his important contributions to the development of perturbation theory, Euler (in the early 1760s) worked on an alternative technique for solving the three-body problem (Eqn (9.2)), which he hoped would shed light on the Sun-Earth-Moon system. If two of the bodies in a three-body system are much more massive than the third, one might assume that the small mass is affected by the other two bodies, but not the other way around. In such a situation, the two massive bodies - usually referred to as primaries - satisfy the equations for a two-body system, and thus move in Keplerian orbits. The problem then is to determine how the small body moves within the resulting gravitational field. Euler made significant advances in the study of this 'restricted three-body problem'; in particular, in 1767, he showed that exact solutions to the equations were possible, representing situations in which the three bodies remained forever in a straight line, the line itself rotating in space.67

The idea of finding exact solutions to the full three-body problem was then pursued by Lagrange. Although having declined to enter the prize contests of 176868 and 1770, Lagrange did enter the 1772 contest for which he shared the prize with Euler. He wrote to d'Alembert:

I have considered the three-body problem in a new and general manner, not that I believe it is better than the one previously employed, but only to approach it [in another way].

Apart from Euler's work discussed above, all attacks on the three-body problem had been based on perturbation theory. Lagrange tried a new approach and succeeded in finding a number of exact solutions.

The three-body problem is a system of nine second-order ordinary differential equations and is thus of order 18. Newton showed that the centre of gravity of any system of mutually interacting particles moves with constant velocity

67 Boccaletti and Pucacco (1996), p. 216.

The prize question of 1768 concerned the Moon. D'Alembert encouraged Lagrange to enter, but Lagrange replied: 'The king [of Prussia] would like me to compete for your prize, because he thinks Euler is working on it; that, it seems to me, is one more reason for me not to work on it' (Itard (1975)). Lagrange never met Euler, but the latter's influence on the former's work was perhaps greater than that of anyone else.

69 Quoted from Itard (1975).

(what we would now call the 'conservation of linear momentum')™ and so the system immediately can be reduced to one of order 12. This can be demonstrated from Eqn (9.2) by multiplying by the equation by m, and summing over i . This results in mi r, = 0,

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