**In short:** The question is how to go from the first equation on page 8, [of this paper](https://pdfs.semanticscholar.org/748d/8a4af0ee6001a25395bac5b7077fc6ac5041.pdf) to the second equation. 


**Some background**
I'm working in optimization and I am currently reading a paper 
[see page 8,Proof of Proposition 2.3.](https://pdfs.semanticscholar.org/748d/8a4af0ee6001a25395bac5b7077fc6ac5041.pdf) where the author uses a somewhat geometric notation which makes it a bit hard for me to grasp:


We are studying an equation $$f'(t) = Tf(t) + V(f(t))=H(f(t))$$ with initial value $f(0).$
Thus, the equation is linear in $T$ and in general nonlinear in $V.$


The idea is to split this up into
$$f'\_1(t) = Tf\_1$$ and 
$$f'\_2(t) = V(f\_2(t))$$
because the variation of constant formula teaches us that (now I use the exponential map $e^{tH}(F)f(0) := F(f(t))$, and analogously for $e^{tT}(F)f\_1(0) := F(f\_1(t))$)


$$e^{tH}(id)f(0) = e^{tT}(id)f(0) + \int\_0^t e^{(t-s)H}(D\_V e^{sT}(id))(f(0)) ds$$
where $D\_V$ is the Lie-derivative with respect to vector field $V$.
You find this variation of constant formula in the above paper.
The author then says, that since this equation determines $exp(tH)$ we can apply it again for the exponential $exp((t-s)H)$ appearing in the integrand. You see this step carried out by the author in the above paper.


However, I do not see why this is true, as this equation above only determines $e^{tH}(id)$ and not $e^{(t-s)H}(D\_V e^{sT}(id))$ which appears in the integrand.


Can anybody shed some light on the arguments of the author of this paper?



Exponential map/ Lie derivative in variation for constant formula for ODE
I think the key is that the variation of constants formula holds for any vector field $G$, i.e. the vector $G\_{\varphi\_H^t(v)}$ can be written as
$$G\_{\varphi\_H^t(v)}=\big(e^{tD\_H}G\big)\_v=\left(e^{tD\_T}G+\int\_0^te^{(t-s)D\_H}D\_Ve^{sD\_T}G\,ds \right)\_v. \tag1$$
Indeed, use the fact from page 7 that
$$\frac{d}{dt}\big(e^{tD\_F}G\big)\_v=\big(D\_Fe^{tD\_F}G\big)\_v$$
in order to verify that 
$$\frac{d}{dt}\left(e^{tD\_T}G+\int\_0^te^{(t-s)D\_H}D\_Ve^{sD\_T}G\,ds \right)\_v=\left(D\_H \left[e^{tD\_T}G+\int\_0^te^{(t-s)D\_H}D\_Ve^{sD\_T}G\,ds \right]\right)\_v.$$


This equation (1) is applied in the first equation on page 8 of the cited paper by setting $G=\text{Id}$ and in the second equation by setting $G=D\_Ve^{sD\_T}\text{Id}$.


