The solutions of time-independent systems with one degree of freedom
can be found by quadrature. Such systems conserve the Hamiltonian:
the Hamiltonian has a constant value on each realizable trajectory. We
can use this constraint to eliminate the momentum in favor of the
coordinate. Thus Hamilton's equations reduce to a single equation
*D**q*(*t*) = *f*(*q*(*t*)). The solution *q* can be expressed as a definite integral.

A geometric view reveals more structure. Time-independent systems with one degree of freedom have a two-dimensional phase space. Energy is conserved, so all orbits are level curves of the Hamiltonian. The possible orbit types are restricted to curves that are contours of a real-valued function. The possible orbits are paths of constant altitude in the mountain range on the phase plane described by the Hamiltonian.

Only a few characteristic features are possible.
There are points that are stable equilibria of the dynamical system.
These are the peaks and pits of the Hamiltonian mountain range. These
equilibria are stable in the sense that neighboring trajectories on
nearby contours stay close to the equilibrium point. There are orbits
that trace simple closed curves on contours that surround a peak or
pit, or perhaps several peaks. There are also trajectories lying
on contours that cross at a saddle point. The crossing point is an
unstable equilibrium, unstable in the sense that neighboring
trajectories leave the vicinity of the equilibrium point. Such
contours that cross at saddle points are called *separatrices* (singular: *separatrix*),
contours that ``separate'' two regions of distinct behavior.

At every point Hamilton's equations give a unique rate of evolution and direct the system to move perpendicular to the gradient of the Hamiltonian. At the peaks, pits, and saddle points, the gradient of the Hamiltonian is zero, so according to Hamilton's equations these are equilibria. At other points, the gradient of the Hamiltonian is nonzero, so according to Hamilton's equations the rate of evolution is nonzero. Trajectories evolve along the contours of the Hamiltonian. Trajectories on simple closed contours periodically trace the contour. At a saddle point, contours cross. The gradient of the Hamiltonian is zero at the saddle point, so a system started at the saddle point does not leave the saddle point. On the separatrix away from the saddle point the gradient of the Hamiltonian is not zero, so trajectories evolve along the contour. Trajectories on the separatrix are asymptotic forward or backward in time to a saddle point. Going forward or backward in time, such trajectories forever approach an unstable equilibrium but never reach it. If the phase space is bounded, asymptotic trajectories that lie on contours of a smooth Hamiltonian are always asymptotic to unstable equilibria at both ends (but they may be different equilibria).

These orbit types are all illustrated by the prototypical phase plane
of the pendulum (see figure 3.4). The
solutions lie on contours of the Hamiltonian. There are three regions
of the phase plane; in each the motion is qualitatively different. In
the central region the pendulum oscillates; above this there is a
region in which the pendulum circulates in one direction; below
the oscillation region the pendulum circulates in the other direction.
In the center of the oscillation region there is a stable equilibrium,
at which the pendulum is hanging motionless. At the boundaries
between these regions, the pendulum is asymptotic to the unstable
equilibrium, at which the pendulum is standing upright.^{18}
There are two asymptotic trajectories, corresponding to the two ways
the equilibrium can be approached. Each of these is also asymptotic
to the unstable equilibrium going backward in time.

^{18} The pendulum has only one unstable equilibrium. Remember
that the coordinate is an angle.