How to graphically represent $\ddot x$? We know that given a differential equation:
$$\dot x = f(x), x \in X$$
The $\dot x$ is understood as the tangent vector on the solution trajectory $x$ lying in the tangent space of $X$

What about eequations of the type $\ddot x = f(x)$? Can we think of $\ddot x$ as the tangent vector to the tangent vector? How do you represent $\ddot x$?

A field of mathematics dedicated problems of this type is Symplectic Geometry. One can define $p=\dot{x}$, giving you the following set of equations
$$\dot{\eta}=\frac{d}{dt}{(x,p)=(p,f(x))}$$
where $t$ represents the trajectory parameter as usual, and $\eta$ is now a special type of $2n$-tuple, where $n=\dim (X)$. 
