5Symmetry methods in PDEs
II Integrable Systems
5.2 Vector fields and one-parameter groups of transforma-
tions
Ultimately, we will be interested in coordinate transformations born of the action
of some Lie group. In other words, we let the Lie group act on our coordinate
space (smoothly), and then use new coordinates
˜
x = g(x),
where
g ∈ G
for some Lie group
G
. For example, if
G
is the group of rotations,
then this gives new coordinates by rotating.
Recall that a vector field
V
:
R
n
→ R
n
defines an integral curve through the
point x via the solution of differential equations
d
dε
˜
x = V(
˜
x),
˜
x(0) = x.
To represent solutions to this problem, we use the flow map g
ε
defined by
˜x(ε) = g
ε
x = x + εV(x) + o(ε).
We call
V
the generator of the flow. This flow map is an example of a one-
parameter group of transformations.
Definition
(One-parameter group of transformations)
.
A smooth map
g
ε
:
R
n
→ R
n
is called a one-parameter group of transformations (1.p.g.t) if
g
0
= id, g
ε
1
g
ε
2
= g
ε
1
+ε
2
.
We say such a one-parameter group of transformations is generated by the vector
field
V(x) =
d
dε
(g
ε
x)
ε=0
.
Conversely, every vector field
V
:
R
n
→ R
n
generates a one-parameter group of
transformations via solutions of
d
dε
˜
x = V(
˜
x),
˜
x(0) = x.
For some absurd reason, differential geometers decided that we should repre-
sent vector fields in a different way. This notation is standard but odd-looking,
and is in many settings more convenient.
Notation.
Consider a vector field
V
= (
V
1
, ··· , V
n
)
T
:
R
n
→ R
n
. This vector
field uniquely defines a differential operator
V = V
1
∂
∂x
1
+ V
2
∂
∂x
2
+ ··· + V
n
∂
∂x
n
.
Conversely, any linear differential operator gives us a vector field like that. We
will confuse a vector field with the associated differential operator, and we think
of the
∂
∂x
i
as a basis for our vector field.
Example. We will write the vector field V = (x
2
+ y, yx) as
V = (x
2
+ y)
∂
∂x
+ yx
∂
∂y
.
One good reason for using this definition is that we have a simple description
of the commutator of two vector fields. Recall that the commutator of two vector
fields V, W was previously defined by
[V, W]
i
=
V ·
∂
∂x
W −
W ·
∂
∂x
V
i
= V
j
∂W
i
∂x
j
− W
j
∂V
i
∂x
j
.
Now if we think of the vector field as a differential operator, then we have
V = V ·
∂
∂x
, W = W ·
∂
∂x
.
The usual definition of commutator would then be
(V W − W V )(f) = V
j
∂
∂x
j
W
i
∂f
∂x
i
− W
j
∂
∂x
j
W
i
∂f
∂x
i
=
V
j
∂W
i
∂x
j
− W
j
∂V
i
∂x
j
∂f
∂x
i
+ V
j
W
i
∂
2
f
∂x
i
∂x
j
− W
j
V
i
∂
2
f
∂x
i
∂x
j
=
V
j
∂W
i
∂x
j
− W
j
∂V
i
∂x
j
∂f
∂x
i
= [V, W] ·
∂
∂x
f.
So with the new notation, we literally have
[V, W ] = V W − W V.
We shall now look at some examples of vector fields and the one-parameter
groups of transformations they generate. In simple cases, it is not hard to find
the correspondence.
Example. Consider a vector field
V = x
∂
∂x
+
∂
∂y
.
This generates a 1-parameter group of transformations via solutions to
d
˜
x
dε
= ˜x,
d˜y
dε
= 1
where
(˜x(0), ˜y(0)) = (x, y).
As we are well-trained with differential equations, we can just write down the
solution
(˜x(ε), ˜y(ε)) = g
ε
(x, y) = (xe
ε
, y + ε)
Example. Consider the natural action of SO(2)
∼
=
S
1
on R
2
via
g
ε
(x, y) = (x cos ε − y sin ε, y cos ε + x sin ε).
We can show that
g
0
=
id
and
g
ε
1
g
ε
2
=
g
ε
1
+ε
2
. The generator of this vector
field is
V =
d˜x
dε
ε=0
∂
∂x
+
d˜y
dε
ε=0
∂
∂y
= −y
∂
∂x
+ x
∂
∂y
.
We can plot this as:
Example. If
V = α
∂
∂x
,
then we have
g
ε
x = x + αε.
This is a translation with constant speed.
If we instead have
V = βx
∂
∂x
,
then we have
g
ε
x = e
βε
x,
which is scaling x up at an exponentially growing rate.
How does this study of one-parameter group of transformations relate to our
study of Lie groups? It turns out the action of Lie groups on
R
n
can be reduced
to the study of one-parameter groups of transformations. If a Lie group
G
acts
on
R
n
, then it might contain many one-parameter groups of transformations.
More precisely, we could find some elements
g
ε
∈ G
depending smoothly on
ε
such that the action of g
ε
on R
n
is a one-parameter group of transformation.
It turns out that Lie groups contain a lot of one-parameter groups of trans-
formations. In general, given any
g
(
t
)
∈ G
(in a neighbourhood of
e ∈ G
), we
can reach it via a sequence of one-parameter group of transformations:
g(t) = g
ε
1
i
1
g
ε
2
i
2
···g
ε
N
i
N
.
So to understand a Lie group, we just have to understand the one-parameter
groups of transformations. And to understand these one-parameter groups, we
just have to understand the vector fields that generate them, i.e. the Lie algebra,
and this is much easier to deal with than a group!