Equation for two particle elastic collision, both having velocity and size, in arbitrarily high dimension Is there any theory about general particle elastic collision (both having velocity, different mass, and different size) in arbitrarily high dimension?
I was thinking about conservation of momentum and energy, but I kind of confused myself. Since given dimension N, there are 2N unknowns, but N+1 equations for momentum and energy separately.
 A: For an elastic collision one could always choose a coordinate system with one of the axes along the line connecting the centers-of-mass of the two bodies. The problem then becomes a one-dimensional one (provided, we can treat the objects as point-like in the collision.)
Not sure though what is meant by partially elastic collision here. Elastic collision means that energy and momentum are conserved - that the objects have different masses doesn't change the nature of the collision.
A: starting with   2 particle collision :
\begin{align*}
&m_1\,(\mathbf v_1-\mathbf u_1)=-\lambda_{12}\,\mathbf{\hat{r}}_{12}\\
&m_2\,(\mathbf v_2-\mathbf u_2)=+\lambda_{12}\,\mathbf{\hat{r}}_{12}\\
&\left[~\mathbf v_2-\mathbf v_1+\epsilon(\mathbf u_2-\mathbf u_1)~\right]\cdot \,\mathbf{\hat{r}}_{12}=0\\
&\text{with}\\
&\mathbf{\hat{r}}_{12}=\frac{\mathbf r_2-\mathbf r_1}{\parallel\mathbf r_2-\mathbf r_1\parallel}
\end{align*}
where $~\mathbf v_i~$ the velocity after the collision,  $~\mathbf u_i~$ the starting velocity and $~\epsilon~$ the restoration coefficient.
$~(\epsilon=1~$ elastic collision).
you obtain 5 equations for the 5 unknowns $~\mathbf v_1~,\mathbf v_2~,\lambda_{12}~$
writing the equations   with matrix notation $~\mathbf A\,\mathbf x=\mathbf b$
\begin{align*}
 &\underbrace{\begin{bmatrix}
    m_1\,\mathbf I_2 & 0\mathbf I_2 &+\mathbf{\hat{r}}_{12}^T \\
    0\mathbf I_2 & m_2\,\mathbf I_2 & -\mathbf{\hat{r}}_{12}^T \\
    -\mathbf{\hat{r}}_{12}^T & +\mathbf{\hat{r}}_{12}^T & 0 \\
  \end{bmatrix}}_{\mathbf A_{5\times 5}}
 \underbrace{ \begin{bmatrix}
    \mathbf v_1 \\
    \mathbf v_2 \\
    \lambda_{12} \\
  \end{bmatrix}}_{\mathbf x}=
  \underbrace{\begin{bmatrix}
    m_1\,\mathbf u_1 \\
    m_2\,\mathbf u_2 \\
    \epsilon\,\mathbf{\hat{r}}_{12}\cdot (\mathbf u_1-\mathbf u_2) \\
  \end{bmatrix}}_{\mathbf b}\tag 1
\end{align*}
you can adapt   equation (1) for collision between any two particle, for example three particle collision, the possible collision are between
particle 1 and 2, 1 and 3 and 2 and 3,for each pair  you have to  solve the matrix equation (1).
