Hello, everyone! I'm working on a simulation of charged particles, and I'm trying to figure out a way to get exact equations which fit the particles' motions. However, I've arrived at math which is very difficult, so I thought that I'd ask for help. Basically, I want to know how to find the exact equations for two particles, and then I'll extend it to many of them.(adsbygoogle = window.adsbygoogle || []).push({});

So, imagine that there are two charged particles, y and Y. They each have the same mass and charge, so I'll ignore those aspects. Just know that the product of their charges and Coulomb's constant divided by their mass is a constant I'll dub k. Each has properties that pertain to it. The capital letters always refer to Y's properties, and the lowercase letters refer to y's. Their positions are known as (a,b) and (A,B). So, I know the following pieces of information, based on the force between the two:

[itex]\frac{d^{2}A}{dt^2}=\frac{k(A-a)}{((A-a)^2+(B-b)^2)^{1.5}}[/itex]

[tex]\frac{d^{2}B}{dt^{2}}=\frac{k(B-b)}{((A-a)^2+(B-b)^2)^{1.5}}[/tex]

[tex]\frac{d^2a}{dt^2}=\frac{k(a-A)}{((A-a)^2+(B-b)^2)^{1.5}}[/tex]

[tex]\frac{d^2b}{dt^2}=\frac{k(b-B)}{((A-a)^2+(B-b)^2)^{1.5}}[/tex]

So, with just two particles, I have four differential equations. I have no idea how to solve it. BTW, I also know the particles' positions and velocities at time t=0, so you don't have to bother with the constants of integration. Just leave those C's as they are and I won't complain. :)

Also, I'd appreciate it if you showed me how to work through the problem. If I'm going to extend it to other problems, I'll need to know how. Thanks!

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# Simultaneous Differential Equations

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