▁▁▁▁ ▁with ▁the ▁integral ▁▁▁▁ ▁corresponding ▁to ▁the ▁object ' s ▁angular ▁momentum . ▁Since ▁▁▁▁ ▁subst itut ing ▁the ▁above ▁conser ved ▁quantities ▁we ▁immediately ▁obtain : ▁▁▁▁ ▁which ▁is ▁the ▁equation ▁of ▁the ▁con ic ▁section ▁( with ▁the ▁origin ▁at ▁the ▁focus ) ▁in ▁ped al ▁coordinates ▁( see ▁ped al ▁equation ). ▁Notice ▁that ▁only ▁ 2 ▁( out ▁of ▁ 4 ▁possible ) ▁conser ved ▁quantities ▁are ▁needed ▁to ▁obtain ▁the ▁shape ▁of ▁the ▁orbit . ▁This ▁is ▁possible ▁since ▁the ▁ped al ▁coordinates ▁do ▁not ▁describe ▁a ▁curve ▁in ▁full ▁detail . ▁They ▁are ▁generally ▁ind ifferent ▁to ▁paramet riz ation ▁and ▁also ▁to ▁a ▁rotation ▁of ▁the ▁curve ▁about ▁the ▁origin — which ▁is ▁an ▁advantage ▁if ▁you ▁care ▁only ▁about ▁the ▁general ▁shape ▁of ▁the ▁curve ▁and ▁do ▁not ▁want ▁to ▁be ▁dist ract ed ▁by ▁details . ▁ ▁This ▁approach ▁can ▁be ▁applied ▁to ▁a ▁wide ▁range ▁of ▁central ▁and ▁Lor ent z - like ▁force ▁problems ▁as ▁discovered ▁by ▁P . ▁Bl as ch ke ▁in ▁ 2 0 1 7 . ▁ ▁See ▁also ▁ ▁Action - angle ▁coordinates ▁ ▁Ber tr and ' s ▁theorem ▁ ▁B inet ▁equation ▁ ▁Hamilton – Jac obi ▁equation ▁ ▁Lap lace – Run ge – L enz ▁vector ▁ ▁Ke pler ▁orbit ▁ ▁Ke pler ▁problem ▁in ▁general ▁relativ ity ▁ ▁Ke pler ' s ▁equation ▁ ▁Ke pler ' s ▁laws ▁of ▁planet ary ▁motion ▁ ▁Ped al ▁equation ▁ ▁References ▁ ▁Category