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 .