
219Harmonic Oscillator
© 2010 Taylor & Francis Group, LLC
and
w = −ω
1
Ae
−βt
sin(ω
1
t + θ).
Using polar coordinates r and ϕ (Figure 7.14a):
ru
=+ =
22
1
,
we see that
rAe=
−
ω
βω φ
1
1
(/ )
which is the equation of a logarithmic spiral. Now, the transformation from (x, y) to (u, w) is linear,
so the form of the phase path in the phase plane
is the same as in the u–w plane. This is shown
in Figure 7.14b, where the phase path is a spiral heading inward toward the asymptotically stable
equilibrium state at the origin (0,0), which is called a point attractor or a focal point.
If β > ω
0
(overdamping), the general solution is given by Equation 7.50, which can ...