3
Two-Dimensional Chapter Phase Flows
We've seen how, for one-dimensional dynamical systems u = f (u), the possibilities in terms of the behavior of the system are in fact quite limited. Starting from an arbitrary initial condition u(0), the phase flow is monotonically toward the first stable fixed point encountered. (That point may lie at infinity.) No oscillations are possible1 . For N = 2 phase flows, a richer set of possibilities arises, as we shall now see.
3.1
3.1.1
Harmonic Oscillator and Pendulum
Simple harmonic oscillator
A one-dimensional harmonic oscillator obeys the equation of motion, m d2 x = -kx , dt2 (3.1)
where m is the mass and k the force constant (of a spring). If we define v = x, this may be written as the N = 2 system, d dt where = known: x v = 0 1 - 2 0 x v = v - 2 x , (3.2)
k/m has the dimensions of frequency (inverse time). The solution is well v0 sin(t) v(t) = v0 cos(t) - x0 sin(t) .
x(t) = x0 cos(t) +
(3.3) (3.4)
1 If phase space itself is multiply connected, e.g. a circle, then the system can oscillate by moving around the circle.
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CHAPTER 3. TWO-DIMENSIONAL PHASE FLOWS
Figure 3.1: Phase curves for the harmonic oscillator. The phase curves are ellipses: x2 (t) + -1 v 2 (t) = C , (3.5)
2 where the constant C = x2 + -1 v0 . A sketch of the phase curves and of the phase flow 0 is shown in Fig. 3.1. Note that the x and v axes have different dimensions. Note also that the origin is a fixed point, however, unlike the N = 1 systems studied in the first lecture, here the phase flow can avoid the fixed points, and oscillations can occur.
Incidentally, eqn. 3.2 is linear, and may be solved by the following method. Write the equation as = M , with = x x and M= 0 1 2 0 - (3.6)
The formal solution to = M is (t) = eM t (0) . (3.7)
What do we mean by the exponential of a matrix? We mean its Taylor series expansion: eM t = I + M t + Note that M2 = = hence M 2k = (- 2 )k I , M 2k+1 = (- 2 )k M . (3.10) 0 1 - 2 0 0 1 - 2 0 = - 2 I , (3.9)
1 2!
M 2 t2 +
1 3!
M 3 t3 + . . . .
(3.8)
- 2 0 0 - 2
3.1. HARMONIC OSCILLATOR AND PENDULUM
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Thus,
Mt
e
=
k=0
1 (- 2 t2 )k I + (2k)!
k=0
1 (- 2 t2 )k M t (2k + 1)!
= cos(t) I + -1 sin(t) M = cos(t) -1 sin(t) - sin(t) cos(t) . (3.11)
Plugging this into eqn. 3.7, we obtain the desired solution. For the damped harmonic oscillator, we have x + 2 x + 2 x = 0 = M= 0 1 2 -2 - . (3.12)
The phase curves then spiral inward to the fixed point at (0, 0).
3.1.2
Pendulum
Next, consider the simple pendulum, composed of a mass point m affixed to a massless rigid rod of length . m2 = -mg sin . This is equivalent to d dt = - 2 sin , (3.14) (3.13)
where = is the angular velocity, and where = oscillations.
g/ is the natural frequency of small
The phase curves for the pendulum are shown in Fig. 3.2. The small oscillations of the pendulum are essentially the same as those of a harmonic oscillator. Indeed, within the small angle approximation, sin , and the pendulum equations of motion are exactly those of the harmonic oscillator. These oscillations are called librations. They involve a back-and-forth motion in real space, and the phase space motion is contractable to a point, in the topological sense. However, if the initial angular velocity is large enough, a qualitatively different kind of motion is observed, whose phase curves are rotations. In this case, the pendulum bob keeps swinging around in the same direction, because, as we'll see in a later lecture, the total energy is sufficiently large. The phase curve which separates these two topologically distinct motions is called a separatrix .
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CHAPTER 3. TWO-DIMENSIONAL PHASE FLOWS
Figure 3.2: Phase curves for the simple pendulum. The separatrix divides phase spac...
CH03_N=2