Setting up equations for colliding ellipsoids?

Question

I can set up some equations for colliding spheres in D dimensions as follows. Let there be two spheres of mass $m$ and $M$ respectively. Let the first sphere have ingoing velocity vector $u$ and outgoing velocity vector $u'$. Similarly the second sphere has ingoing momenum $v$ and outgoing momentum $v'$. The balls collide tangent to an imagined hyperplane with normal $n$.

The equations are:

(1) Conservation of energy (1 equation): $$mu^2+Mv^2 = mu'^2+Mv'^2$$ (2) Conservation of momentum in direction of normal (1 equation): $$n.(mu+Mv) = n.(mu'+Mv')$$ (3) Momementum is conserved perpendicular to collision normal independently for each sphere (2D-2 equations): $$|n|^2(u-u') = n.(u-u') n$$ $$|n|^2(v-v') = n.(v-v') n$$

This gives 2D equations in which to find the pair of vectors $u'$ and $v'$.

I am trying to set up similar equations for a pair of colliding ellipsoids in D dimensions. This will involve having an antisymmetric angular momentum tensor $\omega^{ij}$ for each ellipsoid. I know the angular momentum must be conserved but not sure how to express this as equations. Additional information needed would be a vector $r$ which equals $p-x$ where $x$ is the center of the ellipsoid at collision and $p$ is the point of impact.

I'm finding it hard to find such a generalised equation in the literature. It is probably not too hard to set up though. (The objects need not be mathematical ellipsoids, just some extended object of some kind).

Answer 1

Collision of two ellipses

• $\mathbf w_i~$ start velocities
• $\mathbf v_i~$ final velocities
• $\omega_i=\frac{d\phi_i}{dt}~$ angular velocities
• $S_i~$ center of masses
• $\mathbf t~$ tangential direction
• $\mathbf n~$ normal direction
• $I_i~$ moment of inertia
• $dp~$ linear momentum

with

$$\mathbf w_i=w_{in}\,\mathbf n+w_{it}\,\mathbf t\\ \mathbf v_i=v_{in}\,\mathbf n+v_{it}\,\mathbf t$$

the equations:

towards the normal direction

$$m_1\left(v_{1n}-w_{1n}\right)=dp\\ m_2\left(v_{2n}-w_{2n}\right)=-dp$$

and for elastic collision $$v_{2n}-v_{1n}=-\left(w_{2n}-w_{1n}\right)$$

towards the tangential direction

$$m_1\left(v_{1t}-w_{1t}\right)=0\\ m_1\left(v_{2t}-w_{2t}\right)=0$$

for the rotations about the center of masses

$$I_1\,\omega_1=dp\,\rho_1\\ I_2\,\omega_2=-dp\,\rho_2$$

you have 7 equations for the 7 unknows

$$ v_{1n}~.v_{2n}~,v_{1t}~,v_{2t}~,dp~,\omega_1~,\omega_2$$

results

$$v_{1n} ={\frac { \left( m_{{1}}-m_{{2}} \right) { w_{1n}}}{m_{{2}}+m_{{1}}}}+2 \,{\frac {m_{{2}}{w_{2n}}}{m_{{2}}+m_{{1}}}}\quad, v_{1t}=w_{1t}\\ v_{2n}={\frac { \left( m_{{2}}-m_{{1}} \right) {w_{2n}}}{m_{{2}}+m_{{1}}}}+2 \,{\frac {m_{{1}}{ w_{1n}}}{m_{{2}}+m_{{1}}}}\quad, v_{2t}=w_{2t}\\ \omega_1=-2\,{\frac {m_{{2}}m_{{1}} \left( -{w_{2n}}+{ w_{1n}} \right) \rho_{{ 1}}}{ \left( m_{{2}}+m_{{1}} \right) I_{{1}}}}\\ \omega_2=2\,{\frac {m_{{2}}m_{{1}} \left( -{w_{2n}}+{ w_{1n}} \right) \rho_{{ 2}}}{ \left( m_{{2}}+m_{{1}} \right) I_{{2}}}}\\ $$