//simple gravity sim
//fun fact: electric forces work similarly, just swap the +=/-=, change mass to charge and divide by the particles' own mass
//this is not-quite-pseudocode, you have to refit it and write some boilerplate
//so no cheap India IT copypasting pls
const double friction = 0.002;
const double gravity = 1.0;
//^ magnitudes for forces
for(int i = 0;i < PLANETS - 1;i++) { //these loops will go over all pairs where i < j
    for(int j = i + 1;j < PLANETS;j++) {
        vector d = planets[j].pos - planets[i].pos; //you need some kind of vector class, obviously
        planets[i].vel += d * gravity * timestep * planets[j].mass / (sqrt(d.x*d.x+d.y*d.y)*(d.x*d.x+d.y*d.y));
        planets[j].vel -= d * gravity * timestep * planets[i].mass / (sqrt(d.x*d.x+d.y*d.y)*(d.x*d.x+d.y*d.y));
        /*look at that mess
          explanation for timestep is below
          gravitational force = g * m1 * m2 / r^2
          because F = m * a we can ignore the particle's own mass
          d.x*d.x+d.y*d.y is equal to the squared magnitude of d, which is a scalar
          but we need a vector, so we multiply by d first, so we have to divide by |d|^3 instead
          btw: be wary of division by zero here
        */
    }
}
for(int i = 0;i < PLANETS;i++) {
    planets[i].pos += planets[i].vel * timestep;
    /*timestep is the amount of time that passes each step
      it can vary if you need variable FPS, but keep in mind
      that simulation accuracy strongly depends on it
      this is Eulerian integration: for fancier methods, look up Verlet or Runge-Kutta
      here, the velocity is multiplied by the square of the timestep
      normally you'd multiply the acceleration once and the velocity once,
      but this will speed up things by at least 0.005%
    */
    planets[i].vel *= pow(1.0 - friction, timestep); //friction, disable if you want long-term simulations to end up horribly inaccurate
}
//and there you have it
//this is easily extendable to 3D or spring forces or whatever you like, too, if you know the (really simple) math
//have fun