infinitesimal generator

Infinitesimal generators of flows

Given a flow on a manifold $M$ we want to know what vector field $X$ could have “made” that flow so the integral curves follow $X$.

Definition

For a flow $\Phi :\mathbb{R}\times M\to M$ (or $\mathcal{D}\to M$), for each $p\in M$ let $X_p={{\Phi }^{(p)}}^{\prime }(0)$ This is the infinitesimal generator of $\Phi$.

Motivation

This makes sense because for a flow, we know that $\Phi (0,p)=p$, and we want ${\Phi }^{(p)}$ to be an integral curve, thus we need the vector field at $p$ $X_p$ to be set equal to the velocity vector of the curve.

Smoothness of generators

If $\Phi :\mathbb{R}\times M\to M$ (or $\mathcal{D}\to M$), is a smooth flow on a smooth manifold $M$ then the infinitesimal generator of $\Phi$ is a smooth vector field.

Proof

todo