Lagrangian formulation 9one particular initial rolling speed in which the ball can just climb to the top of the
hill and come to rest there, as is shown in Fig. 1.4(c). Such a trajectory clearly divides
the phase space between the two types of motion shown in Figs. 1.4(b) and 1.4(d) and
is known as aseparatrix. If this example were extended to include a large number of
hills with possibly different heights, then the phase space would contain a very large
number of separatrices. Such an example is paradigmatic of the force laws that one
encounters in complex problems such as protein folding, one of the most challenging
computational problems in biophysics.
Visualizing the trajectory of a complex many-particle system in phase space is not
possible due to the high dimensionality of the space. Moreover, the phase space may
be bounded in some directions and unbounded in others. For formalpurposes, it is
often useful to think of an illustrative phase space plot, in which some particular
set of coordinates of special interest are shown collectively on oneaxis and their
corresponding momenta are shown on the other with a schematic representation of
a phase space trajectory. This technique has been used to visualize the phase space of
chemical reactions in an excellent treatise by De Leonet al.(1991). In other instances,
it is instructive to consider a particular cut or surface in a large phase space that
represents a set of variables of interest. Such a cut is known as aPoincar ́e section
after the French mathematician Henri Poincar ́e (1854–1912), who, among other things,
contributed substantially to our modern theory of dynamical systems. In this case, the
values of the remaining variables will be fixed at the values they take at the location
of this section. The concept of a Poincar ́e section is illustrated in Fig. 1.5.
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Fig. 1.5A Poincar ́e section. The dark line represents a trajectory,and the collection of
points at which it crosses the plane is the Poincar ́e section.