Hamiltonian Mechanics

The right side of the equations involves all of the terms
∂H
∂ξ A
=
( ∂H
, ∂H )
∂x i ∂p j
but there is a difference of a minus sign between the two equations and the interchange of x i and p i . We
handle this by introducing a matrix called the symplectic form,
( ) 0 1
Ω AB =
−1 0
where
[1] ij
= δ ij
is the n × n identity matrix. Then, using the summation convention, Hamilton’s equations take the form of
a single expression,
∂H
˙ξ A = Ω AB
∂ξ B
We may check this by writing it out explicitly,
(
ẋi
ṗ i ′
)
=
=
=
( 0 δij ′
−δ i ′ j 0
( ∂H δij ′
∂p j ′
−δ i′ j ∂H
(
∂H
∂p i
− ∂H
∂x i ′
∂x j
)
)
) ( ∂H
∂x j
∂H
∂p j ′
In the example above, we have ξ 1 = θ 1 , ξ 2 = θ 2 , ξ 3 = p 1 and ξ 4 = p 2 . In terms of these, the Hamiltonian
may be written as
)
and with
Hamilton’s equations are
⎛ ⎞ ⎛
˙ξ 1
⎜
˙ξ 2
⎟
⎝ ˙ξ 3
⎠ = ⎜
⎝
˙ξ 4
⎛
=
=
⎜
⎝
⎛
⎜
⎝
H = 1 2 H ABξ A ξ B
⎛
H AB =
⎜
⎝
kl 2 + mgl −kl 2 0 0
−kl 2 kl 2 + mgl 0 0
0 0
1
ml 2 0
0 0 0
1
ml 2
∂
∂ξ C
H = 1 2 H ABδ AC ξ B + 1 2 H ABξ A δ BC = H CB ξ B
0 0 1 0
0 0 0 1
−1 0 0 0
0 −1 0 0
⎞ ⎛
⎟ ⎜
⎠ ⎝
0 0
1
ml 2 0
1
0 0 0
ml 2
−kl 2 − mgl kl 2 0 0
kl 2 −kl 2 − mgl 0 0
1
ml 2 ξ 3
1
ml 2 ξ 4
−kl 2 ξ 1 − mglξ 1 + kl 2 ξ 2
−kl 2 ξ 2 − mglξ 2 + kl 2 ξ 1
⎞
⎟
⎠
kl 2 + mgl −kl 2 0 0
−kl 2 kl 2 + mgl 0 0
1
0 0
ml
0
2 1
0 0 0
⎞
⎟
⎠
⎞ ⎛
⎟ ⎜
⎠ ⎝
ξ 1
ξ 2
ξ 3
ξ 4
⎞
⎟
⎠
ml 2
⎞ ⎛
⎟ ⎜
⎠ ⎝
ξ 1
ξ 2
ξ 3
ξ 4
⎞
⎟
⎠
8