# What is the definition of the vector field which is generated by rotations of the circle

I'm reading the first lines of page 5 in the article which called "Cohomologie équivariante et théorème de Stokes", which says

Let $$M$$ be a smooth manifold on which $$S^1$$ acts. Let $$J$$ be the vector field generated by the rotations of a circle.

We denote by $$\mathcal{L}(J)$$ the lie derivative in the direction of $$J$$.

If $$\xi$$ is a vector field which vanishes at a point $$p$$, then $$\mathcal{L}(J)$$ induces an invertible transformation on $$T_pM$$.

What does it mean that $$J$$ is the vector field generated by rotations of a circle?

Given a point $$p \in M$$, the action of $$\mathbb{S}^1$$ yields a path $$\gamma_p(t) = e^{it}\cdot p$$ which is a loop in $$M$$ with basepoint $$p$$. It induces a vector field $$J(p) = \frac{d}{dt}|_{t=0} \gamma_p(t) = \gamma_p'(0)$$.
Edit after reading the article online, it seems you are actually talking about page 6 (not 5) and just a few words after the first sentence "Soit $$M$$ une variété$$\ldots$$", the precise definition of $$J$$ is stated. It is opposite to my answer: the author's choice is $$J_x = -\gamma'_x(0)$$, which is detailed by its action on smooth functions.