674
G Hamiltonian Mechanics
Now we note that because the Hamiltonian is cyclic in Q, its conjugate momentum
P must be a constant which, when written in the form
P =
E
ω
,
allows us to determine the equation of motion for Q as
˙
Q =
∂H
∂P
= ω .
We may now solve this equation for Q(t) to obtain
Q(t) = ωt + δ .
The quantity δ (also called the phase factor) is a constant of integration determined
by the initial conditions. We may now put all of this information together to obtain
q(t) as
q(t) =
2E
mω 2 sin(ωt + δ) .
(G.3.19)
Note that if we set up the phase space representation of a 1-D SHO, we have
p(t) =
√
2P mω cos Q =
√
2mE cos Q =
√
2mE cos(ωt + δ)
≡ b cos(ωt + δ) ,
q(t) =
2E
mω 2 sin(ωt + δ) ≡ a sin(ωt + δ) ,
from which we may write equivalently,
q
a
2 +
p
b
2 = 1 .
Thus, the state of a 1-D SHO at time t lies somewhere on an ellipse in -
space, so that the phase space trajectory for the 1-D SHO forms an ellipse in the
two-dimensional phase space associated with its motion: in the three-dimensional
time-augmented phase space , the trajectory for 1-D SHM is a spiral along the
t-axis, with a projection onto the 2D phase space which is the ellipse appearing
in Fig. G.2.