Lie series

It is a generalization of Taylor series. See @olver86 page 30.
In the context of the note exponential map#Motivation, we have that a vector field $X$ on a manifold $M$ gives rise to a flow in the sense that for a point $p \in M$ and a fixed "time" $t$ we obtain a point $q$ given by

q = p e^{t X} = p lim_{n \to \infty} (I d + t X / n)^{n} .

The velocities of this curves are precisely $X$ .
Now, given a smooth function $F$ defined on $M$ , we observe that if $X = \sum ξ_{i} \partial_{x_{i}}$

\frac{d}{d t} F (p e^{t X}) = \sum_{i = 1}^{n} \partial_{x_{i}} F (p e^{t X}) \cdot ξ_{i} (p e^{t X})

and for $t = 0$

{\frac{d}{d t} |}_{t = 0} F (p e^{t X}) = \sum_{i = 1}^{n} \partial_{x_{i}} F (p) \cdot ξ_{i} (p) = X (F) (p) .

This can be written in another way, according to Taylor

F (p e^{t X}) \approx F (p) + t X (F) (p) $ $ a n d, m o r e o v e r,

F(pe^{tX})\approx F(p)+tX(F)(p)+\frac{t^2}{2}X^2(F)(p)+\frac{t^3}{3!}X^3(F)(p)+\cdots

w h e r e $ X^{k} (F) = X (X (\dots (F))) $ . T h i s e x p r e s s i o n i s c a l l e d L i e s e r i e s . I f w e t a k e $ F = (x_{1}, \dots, x_{n}) $ t h e c o o r d i n a t e f u n c t i o n s, t h e n

pe^{tX}=p+t\xi(p)+\frac{t^2}{2}X(\xi)(p)+\frac{t^3}{3!}X(X(\xi))(p)+\cdots \tag

w h e r e $ ξ = (ξ_{1}, \dots, ξ_{n}) $ a r e t h e c o m p o n e n t s o f $ X $, a n d $ X (ξ) = (X (ξ_{1}), \dots, X (ξ_{n})) $ a n d s o o n . I n t h i s s e n s e, e x p r e s s i o n $ (*) $ i s a " f o r m a l " p o w e r s e r i e s s o l u t i o n t o t h e [[♾ ️ C O N C E P T S / D E s / s y s t e m o f f i r s t o r d e r O D E s ∥ s y s t e m o f f i r s t o r d e r O D E s]]

\dot{x}=X(x).