One parameter groups of transformations, vector fields and differential operators

Symmetries are related to the notions of vector fields and hence differential operators. In this video, we examine the simplest case where this happens, namely, 1-parameter groups of transformations. We start by defining this using the notion of a smooth action of the additive group of real numbers on a manifold. By considering the orbits as trajectories, we show that the resulting velocity vectors naturally define an associated vector field which may be thought of as an infinitesimal generator for the group action. This generator is useful for studying the action, and we give an example how global invariance of the group action can be checked infinitesimally using the infinitesimal generator. Finally, we look at the converse construction, and show how you can almost get a 1-parameter group of transformations from a vector field.