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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 $\vec{r}$ ,$\vec{dl}_r$ and one along the direction of increasing angle $\theta$, $\vec{dl}_{\theta}$ .
  • then i need to find the forces along the given directional vectors.
  • finally i relate $\vec{dl}_r , \vec{dl}_{\theta},\vec{F}_r \mbox{&} \vec{F}_{\theta}$
  • 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.

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The general answer is "Keep studying classical mechanics and the related math.", but that doesn't help you in the means time. The iterative, short step approach your playing around with the beginning of the road for computation methods. Alas, that is also a pretty long road and there is a lot of math--perhaps not as deep as analytic mechanics, but probably more tedious. –  dmckee May 14 '12 at 17:52

1 Answer 1

up vote 3 down vote accepted

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 $\epsilon$, and you add $\epsilon a$ to $v$, and $\epsilon v$ to x, and repeat. Once you find the solution, you make $\epsilon$ 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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What about just two charges Like here the trajectories look simple enough –  The-Ever-Kid Jun 4 '12 at 15:11
@The-Ever-Kid: Those are not trajectories! Those are field lines. These would only be trajectories of a third charged particle in a very dense medium, like honey. The particles have inertia, and there is no better way than simulations. For particles-in-honey you can make pictures like field lines and get the qualitative behavior out. –  Ron Maimon Jun 4 '12 at 18:12
Wait a sec i know what trajectories are and what field lines are here see for your self...You didn't click play.... –  The-Ever-Kid Jun 4 '12 at 19:05
@The-Ever-Kid: It's called "being old" and it's not amazing, it's actually detrimental to doing original thinking. –  Ron Maimon Jun 5 '12 at 0:07
For math, Lang's book on calculus and algebra are good, and people like Hatcher's algebraic topology book (it's available online but I can't say anything from experience). much of William Thurston's geometry work is self-contained, but the nightmare is trying to get Groethendieck's stuff--- which he refuses to allow to be translated and put online. This means you need Mumford's red book (which I don't have, unfortunately, so I'm very stupid here, and I'm very unhappy about it). Milnor's books, and anything by Terrence Tao. It always is nice to read Galileo, Archimedes too, to remind the past. –  Ron Maimon Jun 5 '12 at 5:30

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