pler ▁problem ▁also ▁conser ves ▁the ▁Lap lace – Run ge – L enz ▁vector , ▁which ▁has ▁since ▁been ▁generalized ▁to ▁include ▁other ▁interactions . ▁ ▁The ▁solution ▁of ▁the ▁Ke pler ▁problem ▁allowed ▁scient ists ▁to ▁show ▁that ▁planet ary ▁motion ▁could ▁be ▁explained ▁entirely ▁by ▁classical ▁mechan ics ▁and ▁Newton ’ s ▁law ▁of ▁gravity ; ▁the ▁scientific ▁explanation ▁of ▁planet ary ▁motion ▁played ▁an ▁important ▁role ▁in ▁us her ing ▁in ▁the ▁En light en ment . ▁ ▁Mathemat ical ▁definition ▁ ▁The ▁central ▁force ▁F ▁that ▁var ies ▁in ▁strength ▁as ▁the ▁inverse ▁square ▁of ▁the ▁distance ▁r ▁between ▁them : ▁ ▁where ▁k ▁is ▁a ▁constant ▁and ▁ ▁represents ▁the ▁unit ▁vector ▁along ▁the ▁line ▁between ▁them . ▁ ▁The ▁force ▁may ▁be ▁either ▁attract ive ▁( k < 0 ) ▁or ▁rep uls ive ▁( k > 0 ). ▁ ▁The ▁corresponding ▁scalar ▁potential ▁( the ▁potential ▁energy ▁of ▁the ▁non - central ▁body ) ▁is : ▁ ▁Solution ▁of ▁the ▁Ke pler ▁problem ▁ ▁The ▁equation ▁of ▁motion ▁for ▁the ▁radius ▁ ▁of ▁a ▁particle ▁ ▁of ▁mass ▁ ▁moving ▁in ▁a ▁central ▁potential ▁ ▁is ▁given ▁by ▁Lag range ' s ▁equations ▁▁ ▁and ▁the ▁angular ▁momentum ▁ ▁is ▁conser ved . ▁ ▁For ▁illustr ation , ▁the ▁first ▁term ▁on ▁the ▁left - hand ▁side ▁is ▁zero ▁for ▁circular ▁or bits , ▁and ▁the ▁applied ▁in wards ▁force ▁ ▁equals ▁the ▁cent rip etal ▁force ▁requirement ▁, ▁as ▁expected . ▁ ▁If ▁L