This question arose when I was doing some work with line integrals over force fields. In these questions, the reader is always given a parametrization of the particle's path (in terms of time) and asked to integrate over this path.
If a particle (of mass 1 for simplicity) is placed in force field $$\vec{F}(x,y)=\begin{bmatrix}F_x=P(x,y) \\ F_y = Q(x,y)\end{bmatrix},$$ and this is the net force (i.e. it judges the particle's acceleration in unit distance per unit time squared), how could I model the particle's path in terms of time (i.e. a parametric path)?