# Finding the Parametric Path of a Particle in a Force Field

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, despite the fact of there existing a force field (in terms of position) which may contradict the path in terms of time.

Thus, my question is:

If a particle (of mass 1 for simplicity) is placed at the origin (with initial velocity $$\vec{0}$$) 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)?

You can use the Newtonian equation of motion:

$$\frac{d^{2}}{dt^{2}}\vec{r}(t) = \frac{1}{m} \vec{F}(x(t), y(t))$$ where $$\vec{r}(t)=(x(t), y(t))$$.

When the force is given, you have to solve this equations. This can be non-trivial, when in the most general case your force components have tboth a x and y dependence, because then you have to deal with coupled differential equations.

For a particle upon which a force field $$\vec{F}$$ is applied, its trajectory $$\mathcal{C}$$ (if the problem is studied in two dimensions) can be parametrized by a real parameter $$t$$ (time), so it is given by $$\vec{r}(t)=(x(t),y(t))$$ with $$t \in [t_i,t_f]$$.

...despite the fact of there existing a force field (in terms of position) which may contradict the path in terms of time.

Even though you parametrize the path, in order to evaluate the line integral you must apply this parametrization to both components of the vector field, so you get the components of the force field in terms of time: \begin{align} F_x&=P(x(t),y(t)) \\ F_y&=Q(x(t),y(t)). \end{align} This way, the line integral is given by \begin{align} \int_{\mathcal{C}} \vec{F}\cdot d\vec{r} &= \int_{\mathcal{C}} P(x,y)dx+Q(x,y)dy \\ &= \int^{t_f}_{t_i} P(x(t),y(t))\frac{dx(t)}{dt}dt + \int^{t_f}_{t_i} Q(x(t),y(t))\frac{dy(t)}{dt}dt. \end{align}

...how could I model the particle's path in terms of time (i.e. a parametric path)?

There is not a unique way to parametrize the path taken by the particle, but for simple trajectories like a circumference or straight lines, it's quite easy. For example, suppose the particle travels about a semi-circunference of radius R anti-clockwise, going from $$(x,y)=(R,0)$$ to $$(x,y)=(-R,0)$$. Then, a valid parametrization of $$\vec{r}$$ is the following: \begin{align} x(t)&=R\cos(t) \\ y(t)&=R\sin(t), \end{align} with $$t \in [0,\pi]$$ (you can evaluate $$t=0$$ and $$t=\pi$$ into the parametrization to make sure that this interval in effect represent the path given). Applying this parametrization to the components of the vector field involved in the problem and differentiating $$x(t)$$ and $$y(t)$$ with respect to the parameter, you can evaluate the integral given above.