After identifying the equilibrium states, our next step is to understand the trajectories in phase space in a few limiting cases. In the unforced, conservative limit, a complete account of the orbit structure is given by integrating the equations of motion by using the chain rule in the form
Applying this identity to equation (3.14)
with 
and A = 0
yields
which can be integrated to give
where h is the constant of integration.
The term on the left-hand side of equation (3.20) is proportional to the kinetic energy, while the term on the right-hand side,
is proportional to the potential energy. Therefore, the constant h is proportional to the total energy of the system, as illustrated in Figure 3.6(a).
Figure 3.6: Potential and phase space for a single-mode string (a,c,e) and
beam (b,d,f).
The phase space is a plot of the position x and the velocity v of all the orbits in the system. In this case, the phase space is a phase plane , and in the unforced conservative limit we find
The last equation allows us to explicitly construct the integral
curves (a plot of v(t) vs. x(t)) in the phase
plane. Each integral curve is labeled by a value of h, and the qualitative
features of the phase plane depend critically on the signs of 
and
K.
If 
, then both and K are positive. A plot of
equation (3.22) for several values of h is given in Figure
3.6(c). If 
, then the integral curve consists of a single point
called a center . When 
, the orbits are closed,
bounded, simply connected curves about the center. Each curve corresponds
to a distinct periodic motion of the system. Going back to the string model
again, we see that the center corresponds to the symmetric equilibrium
state of the string,
while
the integral curves about the center correspond to finite-amplitude periodic
oscillations about this equilibrium point.
If 
, then K is negative. The phase plane
has three stationary points.
This parameter regime models a compressed beam .
The left and right stationary
points, 
, are centers, but the unstable point at x = 0, labeled S in
Figure 3.6(b), is a saddle point because it
corresponds to a local maximum of V(x).
Curves that pass through a saddle point are very important and are called separatrices . In Figure 3.6(d) we see that there are two integral curves approaching the saddle point S and two integral curves departing from S. These separatrices ``separate'' the phase plane into two distinct regions. Each integral curve inside the separatrices goes around one center, and hence corresponds to an asymmetric periodic oscillation about either the left or the right center, but not both. The integral curves outside the separatrices go around all three stationary points and correspond to large-amplitude symmetric periodic orbits (Figure 3.6(d)). Thus, the separatrices act like barriers in phase space separating motions that are qualitatively different.
