Question

If you are given a vector field F(x,y) and you drop a particle in the field at point p = (x,y), how would you calculate the parametric curve that the particle takes?

Answer 1

It is called "Integrating equations of motion"
Linear equation of second order. Usually numerically - in arbitrarily genera F

Answer 2

I know what you hoped to obtain @wikiemol , kind of lines of field. But the body does NOT move along them because it has finite mass. It accelerates and overshoots so to speak

Answer 3

So how would you go about "Integrating equations of motion"? I know if you are given a curve and a field you can do a line integral to figure out the work over that field. However i'm sure there are certain vector fields that cause particles to move in a nonlinear path. How would you fine this path given its mass?

Answer 4

depends on the field and the particle...
this is really more of a physics question I think

Answer 5

non charged particles have one formula in a gravitational field, electrically charged experience a force either parallel to motion (on which you can do the line integral to find change in U, but what is the point of that here?) if it is an electric field, or perpendicular to motion if it is magnetic...
I feel like you already knew most of that, so I don't get the question I guess

Answer 6

Least action integral. Let me find you the 'famous' lecture (series)

Answer 7

susskind?

Answer 8

Basically people do all kinds of deepest tricks to get ANALYTICAL and TOPOLOGICAL description of such trajectories. See V, Arnold "DIFFERENTIAL EQUATIONS etc" BOOK THE BEST and MOST CLEAR EXPALANTION OF THE MEANING AND PRINCIPLES OF THIS AREA

Answer 9

http://www.youtube.com/watch?v=h96SW0PfQcg&feature=relmfu

Answer 10

@TuringTest yep:)

Answer 11

I MEAN CLASSICAL MECHANICS EQUATION OF MOTION, IN FORCE FIELDS and such staff as you've asked

Answer 12

http://www.amazon.com/Ordinary-Differential-Equations-Universitext-Vladimir/dp/3540345639/ref=sr_1_1?s=books&ie=UTF8&qid=1348349135&sr=1-1&keywords=arnold+differential

Answer 13

better to start here actually:
http://www.youtube.com/watch?v=3YARPNZrcIY&feature=relmfu

Answer 14

I agree, yet in the generality he asked - the most deep and wide look for later is Arnold

Answer 15

Thanks guys! I happen to be just starting lecture 2 of that very same lecture series @Algebraic! How convenient. @Mikael I will definitely look at that book in greater detail. Thanks also @TuringTest for your input.