My quetion is about the proof of Noether's theorem in Arnold Mathematical Methods Of Classical Mechanics.
Let $M$ be a smooth manifold, suppose the Lagrange function $L(q,\dot{q})$ is a smooth function on the tangent bundle $TM$.
Let $h^s:M\to M,s\in\mathbb{R}$ be a one-parameter group of diffeomorphism that satisfies $$L(h^s_*v)=L(v),\forall v\in TM, \forall s\in\mathbb{R},$$ where $h_*$ is the pushforward induced by $h$.
Let $\varphi(t)$ be a solution of lagrange equation $\frac{\partial L}{\partial q}-\frac{\mathrm d}{\mathrm d t}\frac{\partial L}{\partial\dot{q}}=0$, and let $\Phi(s,t)=h^s(\varphi(t))$.
It seems that Arnold used $\frac{\mathrm d}{\mathrm d s}\frac{\mathrm d}{\mathrm d t}\Phi=\frac{\mathrm d}{\mathrm d t}\frac{\mathrm d}{\mathrm d s}\Phi$ in the proof, but I don't know why does it holds.