Once I asked my teacher how to find the trajectory of any particle that is acted

Question

Once I asked my teacher how to find the trajectory of any particle that is acted upon any force.(Generally)

He Told me that I couldn't do it as I did not know polar coordinate geometry as of then but now I've finally realized the effectiveness of polar coordinates and can solve simple polar geometry problems.Like finding the polar equations for electric equipotentials or the polar equation of electric field lines etc etc.

Now I want to know how to use polar coordinates to find the trajectory of any given particle acted upon by a set amount of varying/steady forces.

For example the trajectory of a particle kept in the vicinity of two charged particles.

I know i begin by selecting the origin .
then i select any assumed trajectory .
then on that trajectory i need to take a small element dl and find its components along the direction of the position vector r? ,dl?r and one along the direction of increasing angle ?, dl?? .
then i need to find the forces along the given directional vectors.
finally i relate dl?r,dl??,F? r&F? ?
finally i get a relation between theta and r which will be the equation of the trajectory.

are my steps correct if not can you guide me to any reference on the net helping me to gain knowledge as to how to go about my problem.

Explanation / Answer

You use a computer--- you pick the initial position and velocity, then you find the force (in x,y coordinates) and therefore the acceleration a, then you pick a small timestep ?, and you add ?a to v, and ?v to x, and repeat. Once you find the solution, you make ? smaller until it stops changing, and this is the answer.

Your teacher is probably incompetently remembering that for any power-law central force, you can solve this in polar coordinates. This is possible using conservation of angular momentum for a central force, and then integrating the radial motion equation directly.

This is useless for more complicated forces, like the many charges you describe. It is impossible to find a solution by any method other than computational approximation. To prove this, note that the particle's position can encode data like in a computer, and an arbitrary force can do arbitrary computations on this data, which means that the only way to figure out what it does is to simulate it directly.

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