Determining the path of a particle in a field

[fishspawned]

Homework Statement

using integration to determine paths of travel

Relevant Equations

F(x,y) = x+y [for example]

This is not a specific homework question, but more of a general query.

If provided with a simple vector field indicating forces (for example, an electrical field), can you use integration to determine the path of a particle placed in that field, if also provided with some initial conditions? Let's say this is a simple 2D plane where the field follows something fairly simple, like F(x,y) = x+ y. Can this be done? Can anyone point me in the right direction to learn more about this?

Consider this as an attempt to change a grid of vectors into a set of field lines that, at the same time, show the path of a set particles within that field

 

[BvU]

A vector field has components. Your F does not.
Apart from that, the answer to your question is yes: you integrate Newton's equations of motion to get the path.
(That's how the SUVAT equations come about).

 

[HallsofIvy]

Your field, F= x+ y, is NOT a "vector field" because x+ y is not a vector!

If you had something like F= xi+ yj then, since Force= mass* acceleration, the acceleration would be $$\frac{d^2x}{dt^2}= x$$ and $$\frac{d^2y}{dt^2}= y$$. The "characteristic equation" of both is $$r^2= 1$$ and So that $$x(t)= Ae^t+ Be^{-t}$$ and $$y(t)= Ce^t+ De^{-t}$$.

Another possibility is that F is not a force vector but a potential energy field. In that case, the force is given by the negative of the gradient of the potential energy field. With potential energy x+ y, the force vector is $$-i- j$$. Then [tex]\frac{d^2x}{dt^2}= -1[tex] and [tex]\frac{d^2y}{dt^2}= -1[tex] so that, integrating twice, $$x(t)= -\frac{t^2}{2}+ At+ B$$ and $$y(t)= -\frac{t^2}{2}+ Ct+ D$$.