Page:Lawhead columbia 0054D 12326.pdf/177

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Fig. 5.1

Now, consider the difference between this case and a case where we start the pendulum at a slightly smaller displacement angle (say, 65 degrees instead of 70). The two trajectories will (of course) start in slightly different places in the state space (both will start at zero angular velocity, but will differ along the other axis). What happens when you let the system run this time? Clearly, the shape it traces out through the state space will look much the same as the shape traced out by the first system: a spiral approaching the point (0,0). Moreover, the two trajectories should never get further apart, but rather will continue to approach each other more and more quickly as they near their point of intersection[1]. The two trajectories are similar enough that it is common to present the phase diagram like Figure 5.1: with just a single trajectory standing in for all the variations. Trajectories which all behave similarly in this way are said to be qualitatively identical. The trajectories for any initial condition like this are sufficiently similar that we simplify things by just letting one trajectory stand in for all the others

  1. This is a defining characteristic of dissipative systems. Conservative systems—undamped pendulums that don’t lose energy to friction—will feature trajectories that remain separate by a constant amount.