Displacement Equation in a Vector Field

Question

While building a program for a college level Physics course, I have a vector field representing force such that $\vec F = F_x(x,y) \hat{i} +F_y(x,y) \hat{j}$. I am asked to create a program which shows the path of a mass $m$ with an initial velocity vector ($\vec v_o$) and position vector ($\vec x_o$) under the influence of the field.

My solution is to make the following steps:

$\vec F = m\frac{\partial^2\vec x}{\partial t^2}$ and as such $\frac{\vec F}{m} = \frac{\partial^2\vec x}{\partial t^2}$

By integration of each side, I find that the path would be

$\vec x = \frac{\vec F \Delta t^2}{2m} +\vec v_o \Delta t +\vec x_o$ Which is the same as the typical displacement equation.

My question is as follows; Is this intuitive use of calculus correct considering that $\vec F$ is position dependent, and thus time-dependent as well?

Answer 1

Note that Newton's equation is not $$\vec{F}=m \frac{\partial^2 \vec{r}}{\partial t^2}$$ But $$\vec{F}=m \frac{\mathrm{d}^2 \vec{r}}{\mathrm{d}t^2}$$ So you have two equations: $$F_x(x(t),y(t))=m \ddot{x}$$ $$F_y(x(t),y(t))=m \ddot{y}$$ So your solution is not correct, because you are treating $F$ as a constant.

Answer 2

Your approach is close to the explicit Euler method. You can also have a look at other numerical methods for ordinary differential equations. Some of them are beneficial if your problem has a particular structure, for instance it is Hamiltonian; in that case symplectic integrators might be preferred.