In Paugams book, Towards the Mathematics of Quantum Field Theory, he writes the following in the introduction:
Let us first illustrate this very general notion by the simple example of Newtonian mechanics in the plane. Let $M=[0,1]$ be the time parameter interval, and $C={\Bbb R}^2$ be the configuration plane. In this case a field is simply a function $x:M \to C$ that represents the motion of a particle in the plane. ...
We will suppose given the additional datum of some starting and ending points $x_0,x_1 \in C$, and of the corresponding speed vectors $\vec{v}_0,\vec{v}_1$...
We may identify the space of trajectories with the space of pairs $(x_0,\vec{v}_0)$ that are initial conditions for the corresponding differential equation. This space of pairs is the cotangent space $T^* C$ of the configuration space.
note:the bar over the x should be a little arrow, and R should be in black-board font, as is usual for the space of real numbers - but I don't know the latex for this. I'll fix it if advised how.
On the face of it, it seems this should be the tangent bundle $TC$ rather than the cotangent bundle $T^*C$. So, why the choice of cotangent bundle here?