Lets do a generic treatment of the problem in 3D space using Newtonian mechanics.
Suppose the vector field in cartesian coordinates is ##\vec{F}(x,y,z,t)=(F_x(x,y,z,t),F_y(x,y,z,t),F_z(x,y,z,t))##. ( I took the field to have 3 components one in each axis and also that the field is time-dependent.).
then from Newton's 2nd law ##\vec{F}=m\frac{d^2\vec{r}}{dt^2}## where ##\vec{r}=(x(t),y(t),z(t))## the position vector of the object at time t, and treating the law component-wise we get the following 3 equations (assuming for simplicity that m=1):
##\frac{d^2x(t)}{dt^2}=F_x(x(t),y(t),z(t),t) (1)##
##\frac{d^2y(t)}{dt^2}=F_y(x(t),y(t),z(t),t) (2)##
##\frac{d^2z(t)}{dt^2}=F_z(x(t),y(t),z(t),t) (3)##
As you can see the 3 differential equations (that are linear regarding their left hand side)
are coupled, meaning for example that x(t) depends also from y(t) and z(t),
and in the right hand side of the equations, that each coordinate appears as argument inside the function of each component of the field , for example in 1st equation x(t) appears inside ##F_x## as its first argument. This fact might make the differential equation non-linear depending what is the form of ##F_x,F_y,F_z## and non-linear differential equations are much harder to solve than linear ones.
These two things make the equations hard to solve even for relatively simple expressions for the components of the field, so quite often we have to resort to numerical solution techniques.
