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.
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.
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).
