▁is ▁not ▁zero ▁the ▁definition ▁of ▁angular ▁momentum ▁allows ▁a ▁change ▁of ▁independent ▁variable ▁from ▁ ▁to ▁▁ ▁giving ▁the ▁new ▁equation ▁of ▁motion ▁that ▁is ▁independent ▁of ▁time ▁▁▁▁▁▁▁▁▁▁▁▁ ▁The ▁expansion ▁of ▁the ▁first ▁term ▁is ▁▁▁▁ ▁This ▁equation ▁becomes ▁qu as il ine ar ▁on ▁making ▁the ▁change ▁of ▁variables ▁ ▁and ▁multip lying ▁both ▁sides ▁by ▁▁ ▁After ▁substitution ▁and ▁re arr ang ement : ▁ ▁For ▁an ▁inverse - square ▁force ▁law ▁such ▁as ▁the ▁gravit ational ▁or ▁electro static ▁potential , ▁the ▁potential ▁can ▁be ▁written ▁▁▁ ▁The ▁orbit ▁ ▁can ▁be ▁derived ▁from ▁the ▁general ▁equation ▁ ▁whose ▁solution ▁is ▁the ▁constant ▁ ▁plus ▁a ▁simple ▁sin us oid ▁ ▁where ▁ ▁( the ▁ecc ent ric ity ) ▁and ▁ ▁( the ▁phase ▁offset ) ▁are ▁constants ▁of ▁integration . ▁ ▁This ▁is ▁the ▁general ▁formula ▁for ▁a ▁con ic ▁section ▁that ▁has ▁one ▁focus ▁at ▁the ▁origin ; ▁▁ ▁corresponds ▁to ▁a ▁circle , ▁ ▁corresponds ▁to ▁an ▁el lipse , ▁ ▁corresponds ▁to ▁a ▁par ab ola , ▁and ▁ ▁corresponds ▁to ▁a ▁hyper bol a . ▁ ▁The ▁ecc ent ric ity ▁ ▁is ▁related ▁to ▁the ▁total ▁energy ▁ ▁( cf . ▁the ▁Lap lace – Run ge – L enz ▁vector ) ▁ ▁Compar ing ▁these ▁formula e ▁shows ▁that ▁ ▁corresponds ▁to ▁an ▁el lipse ▁( all ▁solutions ▁which ▁are ▁closed ▁or bits ▁are ▁elli ps es ), ▁ ▁corresponds ▁to ▁a ▁par ab ola , ▁and ▁ ▁corresponds ▁to