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In my notes, the lecturer considers a smooth vector field $v: TM\to T(TM)$, with $M$ a smooth manifold. Let's write

$$v(u,e)=((u,e), (a(u,e),b(u,e)).$$

It is said that $v$ is a second order equation if $T\pi_M\circ v=\text{id}$. It implies that $a(u,e)=e$, i.e.

$$v(u,e)=((u,e),(e,b(u,e))).$$

Here, my notes claim that if $c(t)=(u(t),e(t))$ is a curve on $TM$ satisfying the above, then

$$\frac{du}{dt}=e(t), \ \frac{de}{dt}=b(t)$$

This is probably very stupid but I can get why...

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