Symmetry groups of the planar 3-body problem and action minimizing trajectories


   
      "!$#&%' )()
* +% ,- ( / .01 223 00 0546078  9$:;0,? @ ACB=AED FHGFJILK=MLNPOEOEQ*R S=T FUB=AEVWNYXZ\[WNPI]I^FJILAEQ*R_F;DVa`bKcLF;DD Fed$NfI]IgFJhPAiDAjR kl ILAEOnm*obSmJp;pJq r s>t8uWvwyxz{|uW}~wt tx € +}~t vƒ‚Wz„uW‚ n… †ˆ‡Š‰U‹HŒgŽƒJ‘5’Ž‡P“”’Ž•—–5‡ŠŽ˜U™š’Ž‡Š‰”–8‡g›+™5œ”Y”Œg‰”Œg‘žLŸ¡  ‡P“P¢£”‘5‡g ”¤†¥$‘_~¤Œ^™š’§¦Š—~¨©˜”’Ž¤’¤ U‘5’¤˜U† †‡g™š’Ž‡Š‰”–?ªy™šœU+~¨©˜”’ŽŒ^™5~‘šŒ^b«>Œg‹g‘šŒg‰”‹gŒg‰U“ ™šœ”•~‡ŠŽ¤’¤‰U¬Œg‘-­˜”¤‘®Ÿ°¯‡Š˜”§™š‡Š‰–5‡ŠŽ˜U™š’Ž‡Š‰”–?ªŒg‘5 “”7±&‰”’§™š~§¢'™5œ”)–_’¤†”¤–_™Œ^‰”“²†ˆ‡g–_™+³©‰”‡µ´|‰*¶Y·þ9óEü®ïöõ ^ùÿñú]ï  ûƒïñÿñò7ø]ù  ôEò7ÿ Ð÷®õ&Œg‘®™š’Œ^¤Ž¢ Œg‰U–_´~‘™5‡ƒ™šœ”-‡Š ~‰¨f˜U~–_™5’¤‡Š‰Ó¾C ‡Š–_~“Å ½` ¢ S¹œ”‰”•~’Ž‰”~‘šÂ´|œU™šœU~‘ ™šœ”’¤‘ †ˆ’Ž‰”’¤†’¤Ë~‘_–=•~‡g’¤‰”•’¤“” ‡Š‘|‰”‡^™¬ÔW›Í‡Š‘?™¡´‡ˆ‡^›b™5œ”~†¾ ® ­ Œg‰”“ ®}¯X° ™šœUŒ^‰”–_´~‘-’¤–™šœU+›Í‡Š¤Ž‡]´|’Ž‰”‹ •‡Š‰”“”’§™š’¤‡g‰”–|Œg‘5ƒ¦g~‘5’§±&~“;Ô °PO9Q ?VU / / Žjë š ™5œ”Ì†ˆ‡g†ˆ‰©™5˜”†‡^›W’¤‰U~‘_™5’Œ+’Ž–†¾ š_†Â «  aG ¶L•~‡Š–¬„¾ š_Â,¥ ãË ÃŠÈ,ÁfÈ1žU¥ ´|œU~‘5` š8’¤–?™šœU/’Ž‰½™š~‹g~‘Œ/™5’¤†7Ÿj“”~ ~‰”“U~‰½™?•‡Š†ˆU¤7Æ©ŸE¦gŒ^¤˜”“›Í˜”‰U•™š’Ž‡ŠX ‰ W9¾ _ –_˜”•1œº™5œ&Œ^‰™ ÙK¶°¾ _| ËYW9¾ _¢Â ²Z¶E¥W´|œ”~‘_¼¾ ²Z¶7 ¶|’Ž–+Œ«>Œg‹Š‘5Œg‰”‹Š)•~‰½™š‘5Œg?•‡Š‰U±&‹Š˜U‘šŒ^™5’¤‡Š‰*¶ U ˜U™™šœ” †‡Š†~‰½™š˜”†‡g›=’Ž‰”~‘®™š’¤Œ+’¤–?±UÆP~“ÓŒg‰”“*¥&˜U™5‡‘_ʐ É&•™5’¤‡Š‰y™šœ”‘5/’¤–|Œ˜”‰”’Ž¨©˜”•~‰½™š‘5Œg •~‡g‰U±&‹Š˜PŸ ‘5Œ^™š’Ž‡Š‰8´|’Ž™šœ 7 Ë 7EL ¶ A ¶\·–_¢f†ˆ†™5‘_¢ˆ‹Š‘5‡g˜”)‡g› ™¡¢f €\Ä´|’§™šœ8Œ^‘5 ”’§™š‘šŒ^‘_¢2†ˆŒg–5–_~–?’Ž–™šœU$¤’Ž‰”n–®¢f†ˆ†7Ÿ ™5‘_¢ Ô$™šœ”2‹Š‘_‡Š˜”’Ž–ÌŒy‹g‘5‡Š˜”‡^›¹‡Š‘_“”~‘/Á8Œg•7™š’Ž‰”‹8 ½¢—Œ)‘5 É&~•7™š’¤‡g‰'‡Š‰ ™5œ”™š’Ž†ˆ•~’¤‘_•~ŽP¹¥= ½¢ Œ‘5Êɔ~•™5’¤‡Š‰Y‡Š‰—™šœ””¤Œg‰”Rʈ¥bŒg‰”“Y™š‘5’§¦f’ŒgŽŽ¢‡Š‰—™šœ”–57™‡g›¹’¤‰”“U7ÆP~–ì\¶ˆ·Œg‹Š‘5Œg‰”‹Š’¤Œg‰'–_‡Š¤˜P™š’¤‡g‰¼•¬Œ^‰”‰”‡g™ JÅŒ †’¤‰”’Ž†+˜”†¶ S¹‡Š‰”–_’¤“”‘-™5œ”œU‡Š†ˆ‡g‹Š‘šŒgUœ”’¤•ƒ†‡g™š’Ž‡Š‰”–?›Í‘_‡Š†"™šœ”­˜”¤‘n•‡Š‰U±&‹g˜”‘šŒ^™5’¤‡Š‰'¾ ´|’Ž™šœ™šœ”Ì™šœU’¤‘5“   ‡P“P¢8’¤‰8™šœU/•~‰½™š‘1 ¾Í┶ýÁg ٠¯ ¾ _¦Â ËB)hÀ  K jÈ Ù ° ¾ _¦Â Ë ô )hÀ  K °È ÙuÓµ¾ _¦Â Ë à ¾Í›Í‡Š‰‘ 1Ñ»Ú àUÈ,¨Á §*Þi]È ) ð àU*È šeÑóŒg‹g‘šŒg‰”‹gƒ•~‡Š‰U±”‹Š˜”‘šŒL™š’¤‡g‰*¶ i

›Í‡g‘5Ì”‘5‡µ¦f’¤‰U‹`¥‘_‡ŠJ‡g–5’Ž™5’¤‡Š‰Y¾CäU¶§Ã]áŠÂ´/‰U~~“™¡´¹‡”‘_~¤’Ž†ˆ’Ž‰&Œg‘®¢Å« †ˆ†ˆŒ^™1ŒP¶ Œ «(xŽ¹C‘
ìLê J
W$J7HKJ@; U D;O U _eQ=@_Á;PWHKQd U HX;  Ò þ QÚHAl?n@FRC=@?BA U PWPKn@dÓ;eab[/YÖHKJ@;—P U •e;?9Tvc ?=/HKO U C@QRcHKQ?=`HXJ U HÂHKJ@; OW;QP U c?FFRQPKQR?=2Q="PWH U =ÇH U HxHKJ@;Ô[j?n"=@C U OYÍHXQRdf; q % – U ×K=@× CmF;E|H - * [j;ÔQÚHXcP a8*!ÔQPW] ?eHXOW? G/YÖ× PWn@[@_OW?n@G a)AxJ@QRc­JÖQRP U _OK?n"G ?eT6?OWC@; OÂHA~? PWQ=@c™ ; -˜œ ‡ - _; =@;O U HK; Cý[/> Y K * T  K * U cEHXP>HKOKQÚDNQ U FFÚY2?=4HKJ@;Q=@C";EL PW;H“HXJ"; =ÕVJ@;?OK;d Ò ! t× F; U C@P>HK? U c ?=/HKO U C@QRcHKQ?= ! :QRHKJ@?n"H>F?PWPÁ?eT±_;=@; O U FQÚHYmAl;c U = HKJ@; =PWn@G@Gb?PK;—HKJ % U "H  K * ×   × ! ü PÂHK? Mb K * × abHXJ@OW; ;c U PK;P U OW;—Gb?PKPWQ[@FRJ; IV; QÚHXJ@;O Mb K *% × QRP~HXJ@;“QRC@; =/HKQRH% YaNHXJ@; U =ÇHKQGb?NC U % F d Ø U GÍ?O U OK;Ÿž@; cHKQ?=2AxQÚHXJ2OK;PKGb; cEH˜HK? U FQR=@;¡  !v] =2HXJ@;ÁÀ@OKPWH HAl?Ic U PK;P˜HKJ@;ÔOK?eH U HXQR=@_c QROKc FR;ÁG@OW?Gj;OWHY`J@?eFC@P U ="CÖPW?ZAl; U OK;ÔF;ET.HSAxQRHKJmHKJ@;ÔF U HWHX;OSc U PW; ! õë;HHeg 5 ?    8A@ [b;ZHXJ@;Óc ?FRFQRC@Q=@_Ic Fn"PWË HX;O k QÚHÔQPÔ; U PWYHX?ÖPK;;ÓHKJ U H; QÚHXJ@;OHe ?   Ø @ ?HO e 5±! õë;QH Ø ¢ q × «§ q ×  % « * q × a{Ax' J@; OW; « * q × QPÔHXJ@;fc ;=ÇHK; O?9TVd U PKP—?9TSHXJ@; [b?NC"Q; PxQR=e^a U =@CmF;EHxF;H’¢£[j;ÔC@;EÀ)=@; CÖ[/Y ¢ £ q × ¤ y   ~  ¢ ¤ q × 7 8

ü PSPKJ@?BAx=2QR= Š ;cHXQR?= Ò ?9T  Ò þ a¶HXJ@;OK; U OK;ÂPK; ÎÇn@;=@c ;P ¤d¥H¦ – PKn"c­JÍHXJ U H¢§£©¨ q × c ?=/D;OK_;P HK? U K * *; ÎÇn@QÚD U OKQ U =/HIG U O U [b?FQRc`dfQR=@QdÓQYQ=@_c?FFRQPKQR?=øHXO U ãW; cEHX?OYa U =@ChHKJ@QPfc ?=/HXO U C@QRcHXP ÕVJ@;?OK;d "! \ × ! ÕVJ@;G"OK?¶?eTöQRPV=@?BA c?dfG"F;HK; ! ™Bš › š.œ/š

ª ¡J&$&«t¬|¡­  þ†®¯¡°²±´³rµ ¶m·¸·º¹¼»’®E½)»|¾À¿BÁÃÂij8·º¹†Å$¶€Æ4¾d·º¹¼»’Çi½ ¾ › ¨  ¦ œ “D ¹ ¦p² · ®  ¦ ´ ¹ ¦ °¹ .´¶µ · ² ¸ ¨fƒ ¸  µ ¨ ¸ ´¶µe ¸ ´ ¹ ª ¹ ® ›\—[ ¹ ! }~QROK•ÇJBU È n"PK; O˜}~?ePWHX?e= ] =@c ! a¤}l?PWHK?= a< ü a  ÏÏ8 ! þ†®¯¡°²±´³rµ ¶m·¸·º¹¼»$®E½)»¾8¿BÁÂ&³À·º¹ÉÅ6¶€Æ~¾Ê·º¹¼»Çi½&j ?e=S* c?FFRQPKQR?=2Gb; OKQR?NC"Qc“PK?eFn"HKQ?=@PxT?O Ø U c F U PKPx?eT§PWY¶dÓdf;EHXOKQRc 8 * [j?¶C"YIHY¶Gb;G"OK?[@FR; dÓP ! ˆ ¦­÷)¦p² šºÌ Ë › ®¯ ¦ œp¹&” ¦ ´ ² .´ ›K¸ ¨|Ì ´ ¸ ² šrÍ a

ÏeÏeoÇ× a  Ï/ÒÎ –/Ò ! Ø  Ø 8 þ†®±º¹:³rÆ~¹¼»HÏi½0»&Ͼ8Ð4Ð4³ÀÆ~¾ »¡ÑĽ0»|¾8¿BÁÓÒQ¶€±º±¸¾ºÔ¸¹:¿B¹¼»ÄÕr½ m

Q=@QRdfQ, U HKQ?=4G"OK?Gb; OHXQ;P?9T ÖÂQRFF)× P§?OK[@QÚHXP U =@C U G@G@FRQc U HXQ?e=@P§HK?PW?dÓ;cØL*[j?¶C"YG"OK?[@FR; dÓP !´Ì ´)´ šdÙ ´ ¹ ® š€Ú“š ¾ ¦ .´²D ¸ ¨6› Û Ì ´ ¸ ² šB” ¦ ´ ƒ .´ › Û ¸  ¨ ›Ôú +V a t Ø –––× a   ÒÎ  t9– ! o þQÜľÀÝB±º¹¼»®E½0»¾8¿BÁßÞf¾8°²¹:¿B³¸à¹–·ºÐm»áQ½âL½f( ;OKQ?¶C@QRc—PK?eFn"HKQ?=@PÂ?eTkÖ U dÓQFÚHX?="Q U =`PWY¶P* HX;dfP“?9T 8 * [j?¶C"YÖHYNGb; !Ì ´)´ šÀÙ ´ ¹ ® šړš ¾ ¦ .´²D ¸ ¨6› Û Ì ´ ¸ ² š$” ¦ ´ ƒ .´ › Û ¸  ¨ ›™• a <  ÏÏ  × a t   Î  o/Ï ! t þQÜľÀ±´ãº·º¶€ÆAÆ~³k»äÇi½)»ä¾8¿8Á ҆¶QcHK?Ocý ! Š  ; [b; J@;FRY ! \ þÂÄÝB¶€¿» !½  ™½ öLNQPHX; ="c ; U =@CdÓQ="QdÓQY  QR=@_G@OW?Gj;OWHKQ; PÂ?eT±OK;EHXOW?_O U C@;—?OW[@QRHKPÂHK?IHXJ@; HXJ"OK; ;©* [b?NCNY2G@OK?e[@F;d AxQRHKJ2D U OKQ?en@PVc­J@?QRc ; P˜?eT§d U PKPK;P a –e–eo ! G@OW; G@OWQ=/H ! Ø Á #¹–¾ »†Å½ ü cHKQ?= Î dfQR=@QdÓQYQ=@_IGj;OKQ?¶C@QRc U ="C Ï þÂÄÝB¶€¿ » !½  ½0»  㠀¾8¿ »HÒh½0»¡¾À¿B ÎÇn U PWQ Î Gb; OWQ?¶C@QcÁPW?FnNHXQ?e=@PSQ=2HKJ@; j * [b?¶C"YÍG@OW?[@FR; dma ––eo ! G@OK;G@OKQR=/H ! Ø  – þÂÄÝB¶€¿BÔ¸¹:¿B¶€±$»’®E½ ü cEHXQ?e=ÌdÓQ="QdÓQY QR=@_ÓPK?FRn"HXQR?=@P“?eTlHXJ@; j ;EAVHX?="Q U =Zå