Hamiltonian Mechanics

where
A =
A −1 =
( √
1 i km 1
2i √ km i √ km −1
(
)
−1 −1
−i √ km i √ km
)
ω =
√
k
m
Therefore, multiplying eq.(1) on the left by A and inserting 1 = A −1 A,
( ) ( ) (
d x
1
dt A = A
m A −1 x
A
p −k
p
)
(2)
we get decoupled equations in the new variables:
( ) ( a q
a † = A
p
⎛
)
= ⎝
(
1
2
1
2
x −
(
x +
)
√ ip
km
)
√ ip
km
⎞
⎠ (3)
The decoupled equations are
or simply
with solutions
( ) (
d a −iω 0
dt a † =
0 iω
ȧ = −iωa
ȧ † = −iωa †
a = a 0 e −iωt
a † = a † 0 eiωt
) ( a
a † )
(4)
The solutions for x and p may be written as
x = x 0 cos ωt + p 0
sin ωt
mω
p = −mωx 0 sin ωt + p 0 cos ωt
Notice that once we specify the initial point in phase space, (x 0 , p 0 ) , the entire solution is determined. This
solution gives a parameterized curve in phase space. To see what curve it is, note that
m 2 ω 2 x 2
2mE + p2
2mE = m 2 ω 2 x 2
p 2 0 + m2 ω 2 x 2 +
0 p 2 0 + m2 ω 2 x 2 0
m 2 ω 2 (
=
p 2 0 + m2 ω 2 x 2 x 0 cos ωt + p ) 2
0
0
mω sin ωt
1
+
p 2 0 + m2 ω 2 x 2 (−mωx 0 sin ωt + p 0 cos ωt) 2
0
m 2 ω 2 x 2 0 p 2 0
=
p 2 0 + m2 ω 2 x 2 +
0 p 2 0 + m2 ω 2 x 2 0
= 1
p 2
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