5.1 Vector Fields on Two-Manifolds Having an Integral 81
FIGURE 5.1.6. Fixed points of the pendulum.
Denoting the first integral of the unforced, undamped Duffing oscillator
by h was meant to be suggestive. The unforced, undamped Duffing oscillator
is actually a Hamiltonian System, i.e., there exists a function h = h(x, y)
such that the vector field is given by
FIGURE 5.1.7. a) Orbits of the pendulum on R2
with φ = ±π identified. b)
Orbits of the pendulum on the cyliner.
82 5. Vector Fields Possessing an Integral
˙x =
∂h
∂y
,
˙y = −
∂h
∂x
(5.1.7)
(we will study these in more detail later). Note that all the solutions
lie on level curves of h which are topologically the same as S1
(or T1
).
This Hamiltonian system is an integrable Hamiltonian system and it has
a characteristic of all n-degree-of-freedom integrable Hamiltonian systems
in that its bounded motions lie on n-dimensional tori or homoclinic and
heteroclinic orbits (see Arnold [1978] or Abraham and Marsden [1978]).
(Note that all one-degree-of-freedom Hamiltonian systems are integrable.)
More information on Hamiltonian vector fields can be found in Chapters
13 and 14.
Example 5.1.2 (The Pendulum). The equation of motion of a simple pendulum
(again, all physical constants are scaled out) is given by
¨φ + sin φ = 0 (5.1.8)
or, written as a system,
˙φ = v,
˙v = − sin φ,
(φ, v) ∈ S1
× R1
. (5.1.9)
This equation has fixed points at (0, 0), (±π, 0), and simple calculations
show that (0, 0) is a center (i.e., the eigenvalues are purely imaginary) and (±π, 0)
are saddles, but since the phase space is the cylinder and not the plane, (±π, 0)
are really the same point (see Figure 5.1.6). (Think of the pendulum as a physical
object and you will see that this is obvious.)
Now, just as in Example 5.1.1, the pendulum is a Hamiltonian system with a
first integral given by
h =
v2
2
− cos φ. (5.1.10)
Again, as in Example 5.1.1, this fact allows the global phase portrait for the
pendulum to be drawn, as shown in Figure 5.1.7a. Alternatively, by gluing the
two lines φ = ±π together, we obtain the orbits on the cylinder as shown in
Figure 5.1.7b.
End of Example 5.1.2
5.2 Two Degree-of-Freedom Hamiltonian Systems
and Geometry
We now give an example of a two degree-of-freedom Hamiltonian system
that very concretely illustrates a number of more advanced concepts that
we will discuss later on.