Calculating the Lie Derivative
It’s all well and good to define the Lie derivative, but it’s not exactly straightforward to calculate it from the definition. For one thing, it requires knowing the flow of the vector field
, which requires solving a differential equation that might be difficult in practice. Luckily, there’s an easier way.
But first, a lemma: if is an interval containing
,
is an open set, and
is a differentiable function with
for all
, then there is another differentiable function
such that
This is basically just like a lemma we proved for functions on star-shaped neighborhoods. Indeed, it suffices to set
Now let ,
, and
is the local flow of a vector field
on a region
containing
. Consider the function
, which satisfies the condition of the above lemma. We can thus write
for some . Or if we write
we can write
We also can see that . And so we can write
So now we calculate
Last time we wrote the action of on a smooth function
as
which pattern we can recognize in our formula. We thus continue
That is, for any two vector fields the Lie derivative is actually the same as the bracket
.