System modeling

• Particles modeling the center of hard spheres.
• • - Spheres can collide with surrounding obstacles
  • - Spheres can collide with each others

• - System: N particles with position \(p_i\), velocity \(v_i\), mass \(m_i\), modeling a sphere of radius \(r_i\).
• • - Initial conditions \(p_i(0)=p_i^0\), \(v_i(0)=v_i^0\)

• - Forces: Single gravity forces \(F_i=m_i\,g\). Collisions handled by impulses.

• - Temporal evolution: Fundamental principle of dynamics \(v_i(t)=p'_i(t)\), \(v'_i(t)=g\)

• - Numerical solution
• • \(\left\{\begin{array}{l}v^{k+1}=v^k + h\, g \\ p^{k+1}=p^{k}+h\,v^{k+1}\end{array}\right.\)