## Hp q 1 p2 1q2 1q q p

where we have set \x = 1 for convenience. The Hamiltonian is a constant of the motion (the total energy) that can be shown to equal 1(e2 - 1), and the curves p2 + 1 /q2 - 2/q = e2 - 1 for the same five values of e are shown on the right of the figure. Each curve represents a solution to the two-body problem with a different total energy, and the arrows indicate the direction the curves are traversed as time increases.

When e = 0 the solution in phase space is simply the point q = 1, p = O.The circular orbit is thus an example of a fixed point. The elliptic orbits correspond to closed curves that surround this fixed point, and the parabolic orbit serves to divide phase space into two regions: one containing the closed periodic orbits, the other occupied by curves representing hyperbolic orbits (not shown). Crucially, the curves in phase space do not intersect orbranchintwo. If they did, at (p0, q0), for example, there would be two possible solutions to the differential equation, both starting from the same state p = p0, q = q0. For a wide class of differential equations, including all those of interest here, it is known that this cannot be the case; given the initial data, the solutions are unique. Knowledge of the location of any fixed points or curves that bound regions of phase space can thus be used to determine qualitatively the nature of all the solutions to a particular differential equation.

In the vicinity of a fixed point, the phase portrait need not take the form of closed curves as it does in Figure 11.2(6) (such a fixed point is called a centre). Poincare realized that, in two dimensions, fixed points of differential equations could, in general, be points to which all nearby solutions tend as time tends to infinity (or alternatively as t ^ -o), called 'nodes', or they could be saddle points, where the nearby trajectories are hyperbolic in nature and resemble map contours in the vicinity of a col.

Unfortunately, the situation usually is much more complex than that shown in Figure 11.2, since only in the most straightforward examples is the phase space two-dimensional. For the planar restricted three-body problem there are two degrees of freedom, but the two position and two momentum variables are related by Jacobi's integral, so the solutions can be represented by curves in a three-dimensional phase space.

This was the problem Poincare studied in his remarkable memoir On the

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