672
G Hamiltonian Mechanics
This last result enables us to rewrite Eq. (G.3.11) as
k
p k −
∂F 1
∂q k
˙
q k −
k
P k +
∂F 1
∂Q k
˙
Q k −
H +
∂F 1
∂t
= −K .
(G.3.12)
Now, because both the q k and Q k coordinates are considered herein to be independent variables, Eq. (G.3.12) can hold identically only if the coefficients of ˙
q k and
˙
Q k vanish separately, which gives
p k =
∂F 1
∂q k
, P k = −
∂F 1
∂Q k
, K = H +
∂F 1
∂t
.
(G.3.13)
The equations for the p k involve only p k , q k , Q k , and t: therefore, we can solve for
the Q k in terms of the p k , q k , and t. Once this has been done, the results thereby
obtained for the Q k in terms of the q k , p k , and t may be employed to obtain the P k
in terms of the q k , p k and t. The second step is often considerably more difficult to
achieve than is the first step in this process. We shall only illustrate the workings of
this procedure via the following example.
Example The 1-D Simple Harmonic Oscillator.
To see how the above procedure works, let us examine how it can be applied
to the description of the well-known one-dimensional simple harmonic oscillator
(SHO) typically studied in first year Physics courses.
For the 1-D SHO, we may treat the linear momentum p and the position x as
generalized coordinates p, q, with the Hamiltonian representing the total energy as
H =
p 2
2m
+
1
2 kq
2 .
We shall identify the ratio of the force constant k and the SHO mass m as ω 2 , the
square of the radial frequency of the SHO, as is traditionally done in treatments of
this problem. This allows us to rewrite the Hamiltonian as
H =
1
2m
(p
2
+ m
2 ω
2 q
2 )
in terms of the generalized coordinates p, q. Because H consists of the sum of
two squares, we shall try for a transformation in which H is cyclic in the new
coordinates: to accomplish this, we shall seek a canonical transformation of the
form
p = f (P ) cos Q , q =
f (P )
mω
sin Q .
The reason for introducing the sine and cosine functions is so that we can utilize the
relation sin
2 Q + cos 2 Q = 1. This proposed transformation then gives rise to the
transformed Hamiltonian
G Hamiltonian Mechanics
This last result enables us to rewrite Eq. (G.3.11) as
k
p k −
∂F 1
∂q k
˙
q k −
k
P k +
∂F 1
∂Q k
˙
Q k −
H +
∂F 1
∂t
= −K .
(G.3.12)
Now, because both the q k and Q k coordinates are considered herein to be independent variables, Eq. (G.3.12) can hold identically only if the coefficients of ˙
q k and
˙
Q k vanish separately, which gives
p k =
∂F 1
∂q k
, P k = −
∂F 1
∂Q k
, K = H +
∂F 1
∂t
.
(G.3.13)
The equations for the p k involve only p k , q k , Q k , and t: therefore, we can solve for
the Q k in terms of the p k , q k , and t. Once this has been done, the results thereby
obtained for the Q k in terms of the q k , p k , and t may be employed to obtain the P k
in terms of the q k , p k and t. The second step is often considerably more difficult to
achieve than is the first step in this process. We shall only illustrate the workings of
this procedure via the following example.
Example The 1-D Simple Harmonic Oscillator.
To see how the above procedure works, let us examine how it can be applied
to the description of the well-known one-dimensional simple harmonic oscillator
(SHO) typically studied in first year Physics courses.
For the 1-D SHO, we may treat the linear momentum p and the position x as
generalized coordinates p, q, with the Hamiltonian representing the total energy as
H =
p 2
2m
+
1
2 kq
2 .
We shall identify the ratio of the force constant k and the SHO mass m as ω 2 , the
square of the radial frequency of the SHO, as is traditionally done in treatments of
this problem. This allows us to rewrite the Hamiltonian as
H =
1
2m
(p
2
+ m
2 ω
2 q
2 )
in terms of the generalized coordinates p, q. Because H consists of the sum of
two squares, we shall try for a transformation in which H is cyclic in the new
coordinates: to accomplish this, we shall seek a canonical transformation of the
form
p = f (P ) cos Q , q =
f (P )
mω
sin Q .
The reason for introducing the sine and cosine functions is so that we can utilize the
relation sin
2 Q + cos 2 Q = 1. This proposed transformation then gives rise to the
transformed Hamiltonian