▁ar ises ▁in ▁many ▁context s , ▁some ▁beyond ▁the ▁physics ▁studied ▁by ▁Ke pler ▁himself . ▁ ▁The ▁Ke pler ▁problem ▁is ▁important ▁in ▁cel est ial ▁mechan ics , ▁since ▁Newton ian ▁gravity ▁ob e ys ▁an ▁inverse ▁square ▁law . ▁ ▁Ex amples ▁include ▁a ▁satellite ▁moving ▁about ▁a ▁planet , ▁a ▁planet ▁about ▁its ▁sun , ▁or ▁two ▁binary ▁stars ▁about ▁each ▁other . ▁ ▁The ▁Ke pler ▁problem ▁is ▁also ▁important ▁in ▁the ▁motion ▁of ▁two ▁charged ▁particles , ▁since ▁C oul omb ’ s ▁law ▁of ▁electro stat ics ▁also ▁ob e ys ▁an ▁inverse ▁square ▁law . ▁ ▁Ex amples ▁include ▁the ▁hydro gen ▁atom , ▁posit ron ium ▁and ▁mu on ium , ▁which ▁have ▁all ▁played ▁important ▁roles ▁as ▁model ▁systems ▁for ▁testing ▁physical ▁theories ▁and ▁meas uring ▁constants ▁of ▁nature . ▁ ▁The ▁Ke pler ▁problem ▁and ▁the ▁simple ▁harm onic ▁oscill ator ▁problem ▁are ▁the ▁two ▁most ▁fundamental ▁problems ▁in ▁classical ▁mechan ics . ▁ ▁They ▁are ▁the ▁only ▁two ▁problems ▁that ▁have ▁closed ▁or bits ▁for ▁every ▁possible ▁set ▁of ▁initial ▁conditions , ▁i . e ., ▁return ▁to ▁their ▁starting ▁point ▁with ▁the ▁same ▁velocity ▁( Ber tr and ' s ▁theorem ). ▁ ▁The ▁Ke pler ▁problem ▁has ▁often ▁been ▁used ▁to ▁develop ▁new ▁methods ▁in ▁classical ▁mechan ics , ▁such ▁as ▁L agr ang ian ▁mechan ics , ▁Hamiltonian ▁mechan ics , ▁the ▁Hamilton – Jac obi ▁equation , ▁and ▁action - angle ▁coordinates . ▁ ▁The ▁Ke