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Let's say I have a vector force field as <x, y>, I.e force depends on the position.

Now I have a body of mass m, initial position S = <p1, p2> and initial velocity u = <u1, u2>. How can I write an equation for the trajectory for this particle with respect to time?

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  • $\begingroup$ Anything that changes the path of a particle from a line with constant velocity is called a force. This is formalized as changing the second derivative from zero, meaning $\ddot p = F(t,p,\dot p)/m$, $p=(x,y)$. So you get a rather general second order differential equation that you sometimes can solve symbolically. Most often the best you get is a function table of approximate values via a numerical method. $\endgroup$
    – Lutz Lehmann
    Commented Oct 18, 2023 at 14:07

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According to Newton's equation of motion, $$\frac{d^2}{dt^2} \bar{r} = \frac{1}{m} \bar{F}(\bar{r})$$

If you have two dimensions, say $\bar{r} = x \cdot \hat{e_x} + y \cdot \hat{e_y}$, then you get two coupled differential equations, $$\frac{d^2}{dt^2} x(t) - \frac{1}{m} F_x(x(t), y(t)) = 0$$ $$\frac{d^2}{dt^2} y(t) - \frac{1}{m} F_y(x(t), y(t)) = 0$$

Depending on the force function, you solve for these differential equations with the initial conditions.

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