<TeXmacs|1.0.0.17> <style|amsart> <\body> Let the overlap area be approximated as <\equation> A(Q<rsub|0>+\<eta\>)=A<rsub|0>+\<eta\>\<cdot\>\<nabla\>A<rsub|0> </equation> where <with|mode|math|\<eta\>> is the correction to the velocities. Minimize the action <\equation> J(\<eta\>)=<big|sum><rsub|i><left|(>\<eta\><rsub|i>m<rsub|i><right|)><rsu\ p|2><right|\|><rsub|f(\<eta\>)=A<rsub|0>+\<eta\>\<cdot\>\<nabla\>A<rsub|0\ >=0> </equation> by setting <\equation> \<nabla\>J(\<eta\>)=\<lambda\>\<nabla\>f(\<eta\>)\<Leftrightarrow\><left|\ {>2\<eta\><rsub|i>m<rsub|i>=\<lambda\>(\<nabla\>A<rsub|0>)<rsub|i><right|\ }>\<Rightarrow\>\<eta\><rsub|i>=<frac|\<lambda\>|2m<rsub|i>>(\<nabla\>A<r\ sub|0>)<rsub|i> </equation> which leads to the equation <\equation> <frac|\<lambda\>|2><big|sum><rsub|i><frac|(\<nabla\>A<rsub|0>)<rsub|i><rs\ up|2>|m<rsub|i>>=-A<rsub|0>\<Rightarrow\>\<lambda\>=<frac|-2A<rsub|0>|<bi\ g|sum><rsub|i>(\<nabla\>A<rsub|0>)<rsub|i><rsup|2>/m<rsub|i>> </equation> and finally <\equation> \<eta\><rsub|i>=-<frac|A<rsub|0>|<big|sum><rsub|j>(\<nabla\>A<rsub|0>)<rs\ up|2><rsub|j>/m<rsub|j>>\<cdot\><frac|(\<nabla\>A<rsub|0>)<rsub|i>|m<rsub\ |i>> </equation> \; With a collision normal <with|mode|math|n>, we require that the new velocities <with|mode|math|<wide|q<rsub|>|\<dot\>><rsub|i><rprime|'>> satisfy <\equation> <wide|q|\<dot\>><rprime|'>\<cdot\>n\<geqslant\>0 </equation> \ by setting <\equation> <wide|q|\<dot\>><rprime|'>=<wide|q|\<dot\>>+\<alpha\>\<eta\><rsub|>\<Righ\ tarrow\><wide|q|\<dot\>>\<cdot\>n+\<alpha\>\<eta\>\<cdot\>n\<geqslant\>0\\ <Rightarrow\>\<alpha\>=-e<frac|<wide|q|\<dot\>>\<cdot\>n|\<eta\>\<cdot\>n\ >,1\<leqslant\>e\<leqslant\>2 </equation> where <with|mode|math|e=1> is a completely inelastic collision, and <with|mode|math|e=2> is a completely elastic collision. \; </body> <\initial> <\collection> <associate|paragraph width|150mm> <associate|odd page margin|30mm> <associate|page right margin|30mm> <associate|page top margin|30mm> <associate|reduction page right margin|25mm> <associate|page type|a4> <associate|reduction page bottom margin|15mm> <associate|even page margin|30mm> <associate|reduction page left margin|25mm> <associate|page bottom margin|30mm> <associate|reduction page top margin|15mm> <associate|language|english> </collection> </initial>