I am trying to prove that if $v=\sum_{i=1}^{n}\zeta_{i}\frac{\partial}{\partial\zeta_{i}}$ is a vector field over the cotangent bundle, $T^{*}(X)$ of a manifold $X$, then its flow in a point $p=(x,\zeta)$ is $p_{t}\colon (x,e^{t}\zeta)$.
In order to do that, I think that I must show that $p_{t}|_{t=0}=p$ (which holds) and that $\frac{d}{dt}(p_{t})=v(p_{t})$.
Can anyone help me understanding how to prove the second one? Thanks in advance.
Let $(U,ψ=(ζ_1,..,ζ_n))$ be a neighbourhoud of $p$ and $ψ$ the chart of the cotagent bundle. Suppose $a(t)$ is an integral curve. Let's call $h(t)=ψ(a(t))=(h_1(t),...,h_n(t))$ the curve mapped into $\mathbb{R^n}$. Now we proceed solving the system of differential equations dictated by the vector field and the point $ψ(ζ)=x=(x_1,...x_n)$ :
$\frac{dh_1(t)}{dt}=h_1(t),h_1(0)=x_1$
$.\\.\\.$
$\frac{dh_n(t)}{dt}=h_n(t),h(0)_n=x_n$
therefore $h_i(t)=x_ie^t$ and so the flow at in $\mathbb{R^n}$ at the point x is $φ_t(x_1,...,x_n)=(x_1e^t,...,x_ne^t)\Rightarrow φ_t(x)=x\cdot e^t$.
Now we have to lift the flow back to the manifold, therefore the flow in $T*(X)$ is:
$Φ_t(ζ)=ψ^{-1}(φ_t(ψ(ζ)))=e^tζ$ . And we are done.