Let's consider many particles $\{X_i=(x_i,y_i,z_i)\}_{i=1}^n \in \mathbb R^{3n} $ with some time-dependent constraints $f(X_i,t)=0$. At each time $t \in \mathbb R $ we have a differential submanifold $M_t=\{\{X_i\}_{i=1}^n \in \mathbb R^{3n}|\; f(X_i,t)=0\}$ and his tangent bundle $TM_t $. Then, you can define two fiber bundles $F=\cup_{t\in \mathbb R}\{t\}\times TM_t$, $G=\cup_{t\in \mathbb R} \{t\}\times TM_t $. A trayectory is a section in $F$, and his velocity a section in $G$. Is that correct mathematically and physically?
1) How can you define a differentiation $F \longrightarrow G$ and in $G$? (I see it physically as the velocity and the acceleration of a trayectory, but I don't understand what it means and how it is mathematically)
2) Suppose the evolution of the particles $X_i(t)$ is determined by a (second oder) ODE's system. Can you define a natural geometry on $F$ and $G$ and a natural geometric concept of "right movement" (or "geodesic movement") that implies the ODE's system? In which situations you can and in which you cannot?
PD: I've read that these problems are related to Jet bundles, but I can't see the connection between both.