
525Hamilton–Jacobi Theory of Dynamics
© 2010 Taylor & Francis Group, LLC
Example 15.3: Simple Harmonic Oscillator (SHO) in Angle and Action Variable
The Hamiltonian of an SHO is
H
p
kq E=+ ==
2
2
1
1
α
in which k is the spring constant of the oscillator. Solving for momentum p, we obtain
pmkq
1
2
α /
from which follows the action variable
Jpqmkq
k== −=
/
42 22
1
2
0
2
1
()
α
π
. (15.34)
Solving for α
1
, we obtain
α
1
1
2
k
m
.
Hence, by Equation 15.31,
w
K
J
k
m
=
∂
∂
1
2π
ν
and we recognize that it is the frequency of the oscillations.
The H-J theory has played a pivotal role in the transition from classical to quantum mechanics.
The first quantum thoughts ...