1Integrability of ODE's
II Integrable Systems
1.3 Canonical transformations
We now come to the main result of the chapter. We will show that we can indeed
integrate up integrable systems. We are going to show that there is a clever
choice of coordinates such Hamilton’s equations become “trivial”. However,
recall that the coordinates in a Hamiltonian system are not arbitrary. We have
somehow “paired up”
q
i
and
p
i
. So we want to only consider coordinate changes
that somehow respect this pairing.
There are many ways we can define what it means to “respect” the pairing.
We will pick a simple definition — we require that it preserves the form of
Hamilton’s equation.
Suppose we had a general coordinate change (q, p) 7→ (Q(q, p), P(q, p)).
Definition
(Canonical transformation)
.
A coordinate change (
q, p
)
7→
(
Q, P
)
is called canonical if it leaves Hamilton’s equations invariant, i.e. the equations
in the original coordinates
˙
q =
∂H
∂q
, p = −
∂H
∂q
.
is equivalent to
˙
Q =
∂
˜
H
∂P
,
˙
P = −
∂
˜
H
∂Q
,
where
˜
H(Q, P) = H(q, p).
If we write x = (q, p) and y = (Q, P), then this is equivalent to asking for
˙
x = J
∂H
∂x
⇐⇒
˙
y = J
∂
˜
H
∂y
.
Example.
If we just swap the
q
and
p
around, then the equations change by a
sign. So this is not a canonical transformation.
Example.
The simplest possible case of a canonical transformation is a linear
transformation. Consider a linear change of coordinates given by
x 7→ y(x) = Ax.
We claim that this is canonical iff AJA
t
= J, i.e. that A is symplectic.
Indeed, by linearity, we have
˙
y = A
˙
x = AJ
∂H
∂x
.
Setting
˜
H(y) = H(x), we have
∂H
∂x
i
=
∂y
j
∂x
i
∂
˜
H(y)
∂y
j
= A
ji
∂
˜
H(y)
∂y
j
=
"
A
T
∂
˜
H
∂y
#
i
.
Putting this back in, we have
˙
y = AJA
T
∂
˜
H
∂y
.
So y 7→ y(x) is canonical iff J = AJA
T
.
What about more general cases? Recall from IB Analysis II that a differ-
entiable map is “locally linear”. Now Hamilton’s equations are purely local
equations, so we might expect the following:
Proposition. A map x 7→ y(x) is canonical iff Dy is symplectic, i.e.
DyJ(Dy)
T
= J.
Indeed, this follows from a simple application of the chain rule.
Generating functions
We now discuss a useful way of producing canonical transformation, known as
generating functions. In general, we can do generating functions in four different
ways, but they are all very similar, so we will just do one that will be useful
later on.
Suppose we have a function
S
:
R
2n
→ R
. We suggestively write its arguments
as S(q, P). We now set
p =
∂S
∂q
, Q =
∂S
∂P
.
By this equation, we mean we write down the first equation, which allows us to
solve for
P
in terms of
q, p
. Then the second equation tells us the value of
Q
in
terms of q, P, hence in terms of p, q.
Usually, the way we use this is that we already have a candidate for what
P
should be. We then try to find a function
S
(
q, P
) such that the first equation
holds. Then the second equation will tell us what the right choice of Q is.
Checking that this indeed gives rise to a canonical transformation is just a
very careful application of the chain rule, which we shall not go into. Instead,
we look at a few examples to see it in action.
Example. Consider the generating function
S(q, P) = q · P.
Then we have
p =
∂S
∂q
= P, Q =
∂S
∂P
= q.
So this generates the identity transformation (Q, P) = (q, p).
Example.
In a 2-dimensional phase space, we consider the generating function
S(q, P ) = qP + q
2
.
Then we have
p =
∂S
∂q
= P + 2q, Q =
∂S
∂P
= q.
So we have the transformation
(Q, P ) = (q, p − 2q).
In matrix form, this is
Q
P
=
1 0
−2 1
q
p
.
To see that this is canonical, we compute
1 0
−2 1
J
1 0
−2 1
T
=
1 0
−2 1
0 1
−1 0
1 −2
0 1
=
0 1
−1 0
So this is indeed a canonical transformation.