store preffered velocity w_alpha separately from the actual v_alpha

now we track w_alpha across steps to preserve speed over time memory (equation 10 and 11)

In the pedestrian 2.pass (integrate) we now integrate dw/dt = F instead of dv/dt = F

Co-authored-by: Copilot <copilot@github.com>

main.cpp

@@ -78,10 +78,12 @@ int main(int argc, const char *argv[]) {
     for (int i = 0; i < n_p; i++) {
       Pedestrian &p = pedestrians[i];
       // # 4 flops = 2 ADDs, 2 MULTs
-      Vector2d w_alpha = p.curr_vel + p.f_i * dt;
+      // integrate preferred velocity:  dw/dt = F           (Eq. 10)
+      p.w = p.w + p.f_i * dt;
+      // Vector2d w_alpha = p.curr_vel + p.f_i * dt; // commented since this was using p.curr_vel instead of p.w (different from paper)
 
       // # 8 flops = 2 MULTs, 2 DIVs, 1 SQRT, 2 MULTs, 1 ADD
-      p.curr_vel = w_alpha * g(p.v_max, w_alpha.norm());
+      p.curr_vel = p.w * g(p.v_max, p.w.norm());
 
       // # 4 flops = 2 ADDs, 2 MULTs
       p.curr_pos = p.curr_pos + p.curr_vel * dt;