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mathematical career with the receipt of his doctorate in 1879. He began teaching at the University of Paris 2 years later. Poincare was endowed with a phenomenal memory and had the capacity to think through difficult problems in his head; only then would he put pen to paper. As a result, his written work was often hastily prepared, requiring corrections and explanations. As soon as

Brief biographical details are given in Goroff (1993), an essay that provides much useful background to Poincare's work.

Poincare

Poincare satisfied himself that he had overcome the fundamental difficulties in a problem, he would move on and leave others to fill-in the details.

Between 1881 and 1886, Poincare published a four-part memoir in which he created the qualitative theory of differential equations. His reasons for approaching such equations this way were stated clearly:

Moreover, this qualitative study has in itself an interest of the first order. Several very important questions of analysis and mechanics reduce to it. Take for example, the three body problem: one can ask if one of the bodies will remain within a certain region of the sky or even if it will move away indefinitely; if the distance between two bodies will infinitely increase or diminish, or even if it will remain within certain limits? Could one not ask a thousand questions of this type which would be resolved when one can construct qualitatively the trajectories of the three bodies? And if one considers a greater number of bodies, what is the question of the invariability of the elements of the planets, if not a real question of qualitative geometry, since to show that the major axis has no secular variations shows that it constantly oscillates between certain limits.

Poincare illustrated the difference between qualitative and quantitative phenomena with reference to the solution of algebraic equations. One can establish the number of real roots without solving the equation - this is qualitative information - but the values of these solutions require a quantitative study.

Hamilton's equations show that in a dynamical system, position and momentum should be considered equally important, in contrast to the more usual equations of motion in which position is treated as the fundamental quantity. This symmetry leads naturally to the concept of phase space, in which the state of a system is given in terms of the positions and momenta of all the constituent particles. Thus, for the n-body problem, the phase space has 6n dimensions. A single point in this 6n-dimensional space defines the state of motion of the entire system.

To illustrate this concept, Figure 11.2 shows solutions to the two-body problem in both real and phase space. On the left are plotted a circular orbit (e = 0), elliptic orbits with eccentricities 0.25, 0.5, and 0.75, and a parabolic orbit (e = 1). In each case, the arrow indicates the direction of motion, the origin corresponds to a focus, and the parameter t = 1, so that the curves have polar equation r = 1/(1 + e cos 0). If we utilize the conservation of angular momentum condition r 20 = constant, the two-body problem can be reduced to a

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