Relative equilibria in the curved N-body problem

_2HiBp2 1[mBHB#`B BM h?2 *m`p2/ N@"Q/v S`Q#H2K

#v

arbM aH2K H?QrBiv

"Xa+X- EBM; #/mHxBx lMBp2`bBiv- kyy3 JXa+X- L2r aQmi? qH2b lMBp2`bBiv- kyRR

:`/mi2 *2`iB}+i2 BM G2`MBM; M/ h2+?BM; BM >B;?2` 1/m+iBQM- lMBp2`bBiv Q7 oB+iQ`B- kyR3

 .Bbb2`iiBQM am#KBii2/ BM S`iBH 6mH}HHK2Mi Q7 i?2 _2[mB`2K2Mib 7Q` i?2 .2;`22 Q7

.P*hP_ P6 S>AGPaPS>u

BM i?2 .2T`iK2Mi Q7 Ji?2KiB+b M/ aiiBbiB+b

Ü arbM aH2K H?QrBiv- kyR3 lMBp2`bBiv Q7 oB+iQ`B

HH `B;?ib `2b2`p2/X h?Bb /Bbb2`iiBQM Kv MQi #2 `2T`Q/m+2/ BM r?QH2 Q` BM T`i- #v T?QiQ+QTvBM; Q` Qi?2` K2Mb- rBi?Qmi i?2 T2`KBbbBQM Q7 i?2 mi?Q`X

BB

_2HiBp2 1[mBHB#`B BM h?2 *m`p2/ N@"Q/v S`Q#H2K

#v

arbM aH2K H?QrBiv

"Xa+X- EBM; #/mHxBx lMBp2`bBiv- kyy3 JXa+X- L2r aQmi? qH2b lMBp2`bBiv- kyRR

:`/mi2 *2`iB}+i2 BM G2`MBM; M/ h2+?BM; BM >B;?2` 1/m+iBQM- lMBp2`bBiv Q7 oB+iQ`B- kyR3

amT2`pBbQ`v *QKKBii22

S`Q7X 1`M2biQ Sû`2x *?p2H- *Q@amT2`pBbQ` U.2T`iK2Mi Q7 Ji?2KiB+b M/ aiiBbiB+bV

S`Q7X aHBK A#`?BK- *Q@amT2`pBbQ`

U.2T`iK2Mi Q7 Ji?2KiB+b M/ aiiBbiB+bV

S`Q7X JB?B aBK- PmibB/2 J2K#2`

BBB

"ah_*h

q2 +QMbB/2` i?2 +m`p2/ N@#Q/v T`Q#H2K- N > 2- QM  bm`7+2 Q7 +QMbiMi :mbbBM +m`pim`2 κ ̸= 0c BX2X- QM bT?2`2b S2

κ- 7Q` κ > 0- M/ QM ?vT2`#QHB+ KMB7QH/b H2κ- 7Q`

κ < 0X Pm` ;QH Bb iQ /2}M2 M/ bim/v `2HiBp2 2[mBHB#`B- r?B+? `2 Q`#Bib r?Qb2 KmimH /BbiM+2b `2KBM +QMbiMi /m`BM; i?2 KQiBQMX q2 }M/ M2r `2HiBp2 2[mBHB#`B BM i?2 +m`p2/ N@#Q/v T`Q#H2K 7Q` N = 4- M/ b22 r?2i?2` #B7m`+iBQMb Q++m` r?2M TbbBM; i?`Qm;? κ = 0X 7i2` Q#iBMBM;  +`Bi2`BQM 7Q` i?2 2tBbi2M+2 Q7 [m/`BHi2`H +QM};m`iBQMb QM i?2 2[miQ` Q7 i?2 bT?2`2- r2 bim/v irQ `2bi`B+i2/ 4@#Q/v T`Q#H2Kb, PM2 BM r?B+? irQ #Q/B2b `2 KbbH2bb - M/ i?2 b2+QM/ BM r?B+? QMHv QM2 #Q/v Bb KbbH2bbX AM i?2 7Q`K2` r2 T`Qp2 i?2 2pB/2M+2 7Q` b[m`2@HBF2 `2HiBp2 2[mBHB#`B- r?2`2b BM i?2 Hii2` r2 /Bb+mbb i?2 2tBbi2M+2 Q7 FBi2@b?T2/ `2HiBp2 2[mBHB#`BX

q2 rBHH 7m`i?2` bim/v i?2 5@#Q/v T`Q#H2K QM bm`7+2b Q7 +QMbiMi +m`pim`2X 6Qm` Q7 i?2 Kbb2b ``M;2/ i i?2 p2`iB+2b Q7  b[m`2- M/ i?2 }7i? Kbb i i?2 MQ`i? TQH2 Q7 S2

κ- r?2M i?2 +m`pim`2 Bb TQbBiBp2- Bi Bb b?QrM i?i `2HiBp2 2[mBHB#`B 2tBbib

r?2M i?2 7Qm` Kbb2b i i?2 p2`iB+2b Q7 i?2 b[m`2 `2 2Bi?2` 2[mH Q` irQ Q7 i?2K `2 BM}MBi2bBKH- bm+? i?i i?2v /Q MQi z2+i i?2 KQiBQM Q7 i?2 `2KBMBM; i?`22 Kbb2bX AM i?2 ?vT2`#QHB+ +b2 H2

κ- κ < 0- i?2`2 2tBbi irQ pHm2b 7Q` i?2 M;mH` p2HQ+Biv r?B+?

T`Q/m+2 M2;iBp2 2HHBTiB+ `2HiBp2 2[mBHB#`B r?2M i?2 Kbb2b i i?2 p2`iB+2b Q7 i?2 b[m`2 `2 2[mHX q2 HbQ b?Qr i?i i?2 b[m`2 Tv`KB/H `2HiBp2 2[mBHB#`B rBi? MQM@2[mH Kbb2b /Q MQi 2tBbi BM H2

κX

"b2/ QM i?2 rQ`F Q7 6HQ`BM .B+m QM i?2 2tBbi2M+2 Q7 `2HiBp2 2[mBHB#`B 7Q` 3@#Q/v T`Q#H2K QM i?2 2[miQ` Q7 S2

κ- r2 BMp2biB;i2 i?2 KQiBQM Q7 KQ`2 i?M i?`22 #Q/B2bX

6m`i?2`KQ`2- r2 bim/v i?2 KQiBQM Q7 i?2 M2;iBp2 +m`p2/ 2@M/ 3@+2Mi`2 T`Q#H2Kb QM i?2 SQBM+`û mTT2` b2KB@THM2 KQ/2HX lbBM; i?Bb KQ/2H- r2 T`Qp2 i?i i?2 2@+2Mi`2 T`Q#H2K Bb BMi2;`#H2- M/ r2 bim/v i?2 /vMKB+b `QmM/ i?2 2[mBHB#`BmK TQBMiX 6m`i?2`- r2 MHvx2 i?2 bBM;mH`BiB2b Q7 i?2 3@+2Mi`2 T`Q#H2K /m2 iQ i?2 +QHHBbBQMc BX2X-i?2 +QM};m`iBQMb 7Q` r?B+? i H2bi irQ #Q/B2b ?p2 B/2MiB+H +QQ`/BMi2bX

Bp

*QMi2Mib

amT2`pBbQ`v *QKKBii22 BB #bi`+i BBB h#H2 Q7 *QMi2Mib Bp GBbi Q7 6B;m`2b pBB +FMQrH2/;2K2Mib pBBB .2/B+iBQM Bt R AMi`Q/m+iBQM R RXR amKK`v M/ Q`;MBxiBQM X X X X X X X X X X X X X X X X X X X X X X X X X X j k 1[miBQMb Q7 JQiBQM Q7 h?2 *m`p2/ L@"Q/v S`Q#H2K 8 kXyXR h?2 TQi2MiBH X X X X X X X X X X X X X X X X X X X X X X X X X X X X X 8 kXyXk 1mH2`Ƕb +HbbB+H i?2Q`2K QM ?QKQ;2M2Qmb 7mM+iBQMb X X X X X X X d kXyXj .2`BpiBQM Q7 i?2 1[miBQMb Q7 JQiBQM X X X X X X X X X X X X X X X 3 kXR 1ti2MbBQM iQ i?2 ~i +b2 X X X X X X X X X X X X X X X X X X X X X X X X X X N kXk "B7m`+iBQM Q7 i?2 BMi2;`Hb Q7 KQiBQM X X X X X X X X X X X X X X X X X X X X RR kXkXR h?2 BMi2;`Hb Q7 i?2 +2Mi`2 Q7 Kbb M/ i?2 HBM2` KQK2MimK X X Rk kXkXk h?2 BMi2;`H Q7 2M2`;v X X X X X X X X X X X X X X X X X X X X X X X X Rk kXkXj h?2 BMi2;`Hb Q7 i?2 iQiH M;mH` KQK2MimK X X X X X X X X X X X Rj

j "B7m`+iBQM Q7 h?2 G;`M;BM P`#Bib R8 jXR _2HiBp2 1[mBHB#`B X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X Rd jXRXR _2HiBp2 2[mBHB#`B 7Q` TQbBiBp2 κ X X X X X X X X X X X X X X X X X X X Rd jXRXk _2HiBp2 2[mBHB#`B 7Q` M2;iBp2 κ X X X X X X X X X X X X X X X X X X Rd jXk h?2 +b2 Q7 TQbBiBp2 +m`pim`2, S2 κ X X X X X X X X X X X X X X X X X X X X X X RN

p

jXkXR _2HiBp2 2[mBHB#`B QM i?2 2[miQ` 7Q` N = 3 X X X X X X X X X X X ky jXkXk _2HiBp2 2[mBHB#`B QM i?2 2[miQ` 7Q` N = 4 X X X X X X X X X X X kR jXkXj 1[mBpH2Mi 2[miBQMb Q7 KQiBQM X X X X X X X X X X X X X X X X X X X kk jXkX9 h?2 +b2 Q7 irQ KbbH2bb #Q/B2b 7Q` N = 4 X X X X X X X X X X X X X kj jXkX8 h?2 +b2 Q7 QM2 KbbH2bb #Q/v 7Q` N = 4 X X X X X X X X X X X X X X kd jXj h?2 +b2 Q7 M2;iBp2 +m`pim`2 7Q` N = 4 QM H2 κ X X X X X X X X X X X X X X jy jXjXR h?2 +b2 Q7 irQ KbbH2bb #Q/B2b QM H2 κ X X X X X X X X X X X X X X X jy jXjXk h?2 +b2 Q7 QM2 KbbH2bb #Q/v QM H2 −1 X X X X X X X X X X X X X X X jR 9 _2HiBp2 1[mBHB#`B BM h?2 *m`p2/ 8@"Q/v S`Q#H2K jj 9XR _2HiBp2 2[mBHB#`B QM i?2 2[miQ` 7Q` 8@#Q/v T`Q#H2K X X X X X X X X X X X jj 9XRXR 1tKTH2 Q7 `2HiBp2 2[mBHB#`B QM i?2 2[miQ` 7Q` 2[mH Kbb2b X X j9 9Xk _2HiBp2 1[mBHB#`B BM i?2 +m`p2/ 8@#Q/v T`Q#H2K BM S2 M/ H2 X X X X X X j8

9XkXR h?2 TQbBiBp2Hv +m`p2/ 8@#Q/v T`Q#H2K rBi? QM2 Kbb i i?2 MQ`i? TQH2 M/ 7Qm` Kbb2b 7Q`K  b[m`2 BM S2 X X X X X X X X X X X X X X jd

9XkXk h?2 M2;iBp2Hv +m`p2/ 8@#Q/v T`Q#H2K rBi? QM2 Kbb i i?2 p2`i2t M/ 7Qm` Kbb2b 7Q`K  b[m`2 BM H2 X X X X X X X X X X X X 9j

8 _2HiBp2 1[mBHB#`B BM h?2 *m`p2/ 6@"Q/v S`Q#H2K 98 8XR _2HiBp2 2[mBHB#`B QM i?2 2[miQ` 7Q` 6@#Q/v T`Q#H2K X X X X X X X X X X X 98 8Xk h?2 6@#Q/v T`Q#H2K rBi? j KbbH2bb #Q/B2b M/ j 2[mH Kbb2b BM S2

κ X X 9e

e h?2 2@M/ 3@*2Mi`2 S`Q#H2Kb QM H2 8y

eXR h?2 2[miBQMb Q7 KQiBQM QM

i?2 SQBM+`û mTT2` b2KB@THM2 KQ/2H X X X X X X X X X X X X X X X X X X X X 8y eXk h?2 M2;iBp2 +m`p2/ irQ +2Mi2` T`Q#H2K X X X X X X X X X X X X X X X X X X 8k eXkXR PM i?2 q2B2`bi`bb KQ/2H L2 X X X X X X X X X X X X X X X X X X X X 8k

eXkXk .vMKB+b `QmM/ i?2 2[mBHB#`BmK TQBMi X X X X X X X X X X X X X X 89 eXkXj "2?pBQ` Q7 i?2 ;HQ#H ~Qr M2` (0, 1, 0, 0)X X X X X X X X X X X X X 88 eXj h?2 i?`22 +2Mi2` T`Q#H2K X X X X X X X X X X X X X X X X X X X X X X X X X X 8e eXjXR h?2 M2;iBp2 +m`p2/ i?`22 +2Mi2` T`Q#H2K X X X X X X X X X X X X X 8d eXjXk h?2 p2`iB+H +b2 X X X X X X X X X X X X X X X X X X X X X X X X X X X 83 eXjXj h?2 ;2Q/2bB+ +b2 X X X X X X X X X X X X X X X X X X X X X X X X X X X 83 eXjX9 *QHHBbBQMb X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X ey

pB

dXR *QM+HmbBQMb X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X eR dXk 6mim`2 qQ`F X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X eR

pBB

GBbi Q7 6B;m`2b

6B;m`2 jXR h?2 +b2 Q7 k 2[mH Kbb2b M/ k KbbH2bb #Q/B2b BM i?2 MQ`i?2`M Q` BM i?2 bQmi?2`M ?2KBbT?2`2 X X X X X X X X X X X X X X X X X X X X X k8 6B;m`2 jXk h?2 +b2 Q7 irQ 2[mH Kbb2b M/ irQ KbbH2bb #Q/B2bX X X X X X X X ke 6B;m`2 jXj  FBi2 +QM};m`iBQM Q7 j 2[mH Kbb2b M/ QM2 KbbH2bb #Q/vX X X X k3 6B;m`2 9XR h?2 +QM};m`iBQM Q7  `2;mH` TQHv;QM Q7 8 2[mH Kbb2bX X X X X X j8 6B;m`2 9Xk h?2 +b2 Q7 4 2[mH Kbb2b M/ QM2 #Q/v i i?2 MQ`i? TQH2 X X X X X jd 6B;m`2 9Xj _2;BQM R1(Γ, z) U+QHQ`2/VX X X X X X X X X X X X X X X X X X X X X X X X 9k

6B;m`2 9X9 h?2 +b2 Q7 4 2[mH Kbb2b rBi? QM2 #Q/v i i?2 p2`i2t BM H2 X X X 9j

6B;m`2 8XR h?2 +b2 Q7 3 2[mH Kbb2b M/ 3 KbbH2bb #Q/B2bX X X X X X X X X X X 9d 6B;m`2 eXR S?b2 bT+2 7Q` y(t), ˙y(t) Ux(t) = 0VX X X X X X X X X X X X X X X X X X 88 6B;m`2 eXk h?2 p2`iB+H +b2 X X X X X X X X X X X X X X X X X X X X X X X X X X X X 83 6B;m`2 eXj h?2 ;2Q/2bB+ +b2 X X X X X X X X X X X X X X X X X X X X X X X X X X X 8N

pBBB

*ELPqG1.:1J1Lha

6B`bi- A rQmH/ HBF2 iQ 2tT`2bb Kv ;`iBim/2 iQ HH? U:Q/V- 7Q` T`QpB/BM; K2 i?2 QTTQ`imMBiv M/ TiB2M+2 iQ +QKTH2i2 Kv S?. bim/vX

a2+QM/- A rQmH/ HBF2 iQ 2tT`2bb Kv bT2+BH i?MF iQ Kv bmT2`pBbQ`bc S`Q72bbQ` 6HQ`BM .B+m- S`Q72bbQ` 1`M2biQ Sû`2x *?p2H -ƘM/ S`Q72bbQ` aHBK A#`?BK 7Q` i?2B` bbBbiM+2- ;mB/M+2- bmTTQ`i- TiB2M+2- `2bT2+i- FBM/M2bb M/ KQiBpiBQMX A K /22THv ;`i27mH iQ i?2B` pBH#BHBiv- ;2M2`Qmb 2M+Qm`;2K2Mib- M/ bm;;2biBQMb Qp2` i?2 Tbi 72r v2`bX h?MF vQm 7Q` ;BpBM; K2 vQm` iBK2 M/ bmTTQ`iX HbQ- A rQmH/ HBF2 iQ i?MF .`X Jm?KK/ a?QB#Ƙ7Q` ?2HT7mH /Bb+mbbBQMbXƘ

AM //BiBQM- i?MFb iQ Kv FBM/ 7i?2` aH2K 7Q` ?Bb HQp2 M/ bmTTQ`iX A /Q MQi ?p2 2MQm;? rQ`/ iQ 2tT`2bb Kv i?MFb iQ ?BKX h?MFb HbQ ;Q2b iQ Kv #2HQp2/ ?mb#M/ E?HB/ 7Q` ?Bb bmTTQ`i- 2M+Qm`;2K2Mi- M/ TiB2M+2 /m`BM; i?2 /m`iBQM Q7 i?Bb S?.X Jv HQp2Hv aQMb U1B/- qHB/- Lr7 M/ P/BV r?Q r2`2 i?2 bQm`+2 Q7 bTB`Bi BM 2p2`v bi2T Q7 Kv S?.X

6m`i?2`- i?MFb ;Q iQ Kv #`Qi?2`b- bBbi2`b- M/ 7`B2M/b 7Q` i?2B` ?2HT M/ bmTTQ`iX Jv S?.X rQ`F rb 7mM/2/ #v i?2 am/B `#BM :Qp2`MK2Mi b+?QH`b?BT- M/ A K p2`v ;`i27mH 7Q` i?2 7mM/BM; QTTQ`imMBivX

6BMHHv- i?MFb iQ HH biz M/ +QHH2;m2b i i?2 .2T`iK2Mi Q7 Ji?2KiB+b BM lMBp2`bBiv Q7 oB+iQ`B 7Q` ?2HTBM; M/ bmTTQ`iBM; K2X

Bt

.1.A*hAPL

A /2/B+i2 Kv rQ`F iQ Kv #2HQp2/ Hi2 KQi?2` LQ`? M/ i?2 /BbiBM;mBb?2/ S`Q72bbQ` 6HQ`BM .B+mX

*?Ti2` R

AMi`Q/m+iBQM

h?Bb /Bbb2`iiBQM /2Hb rBi? `2HiBp2 2[mBHB#`B BM i?2 +m`p2/ N@#Q/v T`Q#H2K Q7 +2H2biBH K2+?MB+b r?2M i?2 T`K2i2` κ +?M;2b bB;M- r?2`2 κ Bb i?2 +QMbiMi :mbbBM +m`pim`2 Q7 i?2 bT+2 BM r?B+? i?2 #Q/B2b `2 KQpBM; Uκ < 0 +Q``2bTQM/b iQ i?2 ?vT2`#QHB+ bT+2- κ = 0 iQ i?2 1m+HB/2M bT+2- M/ κ > 0 iQ i?2 2HHBTiB+ bT+2VX

h?2 +HbbB+H N@#Q/v T`Q#H2K ?b  HQM; ?BbiQ`vX Ab+ L2riQM }`bi T`QTQb2/ Bi BM Re3d BM ?Bb }`bi 2/BiBQM Q7 S`BM+BTB BM i?2 +QMi2ti Q7 i?2 KQQMǶb KQiBQMX >2 bbmK2/ i?i mMBp2`bH ;`pBiiBQM +ib #2ir22M +2H2biBH #Q/B2b U`2/m+2/ iQ TQBMi Kbb2bV BM /B`2+i T`QTQ`iBQM rBi? i?2 T`Q/m+i Q7 i?2 Kbb2b M/ BM BMp2`b2 T`QTQ`iBQM rBi? i?2 b[m`2 Q7 i?2 /BbiM+2X h?2 bim/v Q7 i?2 N@#Q/v T`Q#H2K rb 7m`i?2` /pM+2/ #v "2`MQmHHBb-G;`M;2- GTH+2- 1mH2`- *m+?v- C+Q#B- .B`B+?H2i- SQBM+`û M/ KMv T?vbB+Bbib M/ Ki?2KiB+BMbX

h?2 B/2 Q7 2ti2M/BM; i?2 ;`pBiiBQMH 7Q`+2 #2ir22M TQBMi Kbb2b iQ bT+2b Q7 +QMbiMi +m`pim`2 Q++m``2/ bQQM 7i2` i?2 /Bb+Qp2`v Q7 ?vT2`#QHB+ ;2QK2i`vX AM i?2 R3jyb- BM/2T2M/2MiHv Q7 2+? Qi?2`- "QHvB M/ GQ#+?2pbFv `2HBx2/ i?i i?2`2 Kmbi #2 M BMiBKi2 +QMM2+iBQM #2ir22M i?2 Hrb Q7 T?vbB+b M/ i?2 ;2QK2i`v Q7 i?2 mMBp2`b2-(9)- (kd)- (kj)X  72r v2`b 2`HB2`- :mbb ?/ BMi2`T`2i2/ L2riQMǶb ;`pBiiBQMH Hr b biiBM; i?i i?2 ii`+iBM; 7Q`+2 #2ir22M #Q/B2b Bb BMp2`b2Hv T`QTQ`iBQMH rBi? i?2 `2 Q7 i?2 bT?2`2 Q7 `/Bmb 2[mH iQ i?2 /BbiM+2 #2ir22M i?2 TQBMi Kbb2b UBX2X T`QTQ`iBQMH iQ 1/r2- r?2`2 r Bb i?2 /BbiM+2VX lbBM; i?Bb B/2- "QHvB M/ GQ#+?2pbFv

bm;;2bi2/ i?i- b?QmH/ bT+2 #2 ?vT2`#QHB+- i?2 ii`+iBM; 7Q`+2 #2ir22M #Q/B2b Kmbi #2 BMp2`b2Hv T`QTQ`iBQMH iQ i?2 ?vT2`#QHB+ `2 Q7 i?2 +Q``2bTQM/BM; ?vT2`#QHB+ bT?2`2 UBX2X T`QTQ`iBQMH iQ 1/ sinh2₍

|κ|1/2r)- r?2`2 r Bb i?2 /BbiM+2 M/ κ < 0 Bb i?2 +m`pim`2

Q7 i?2 ?vT2`#QHB+ bT+2VX h?Bb Bb 2[mBpH2Mi iQ bvBM; i?i- BM ?vT2`#QHB+ bT+2- i?2 TQi2MiBH i?i /2b+`B#2b i?2 ;`pBiiBQMH 7Q`+2 Bb T`QTQ`iBQMH iQ coth(|κ|1/2r)X

k

h?2 #Qp2 MHviB+ 2tT`2bbBQM Q7 i?2 TQi2MiBH rb }`bi BMi`Q/m+2/ #v a+?2`BM;-(jj)- (j9)- M/ i?2M 2ti2M/2/ iQ 2HHBTiB+ bT+2 #v EBHHBM;- (RN)- (ky)- (kR)X "mi rBi? MQ rvb Q7 +?2+FBM; i?2 pHB/Biv Q7 i?Bb ;2M2`HBxiBQM Q7 i?2 ;`pBiiBQMH 7Q`+2- Bi rb mM+H2` r?2i?2` i?2 +QiM;2Mi TQi2MiBH ?/ Mv T?vbB+H K2MBM;- i?2 KQ`2 bQ bBM+2 GBTb+?Bix ?/ T`QTQb2/  /Bz2`2Mi 2ti2MbBQM Q7 i?2 Hr- r?B+? im`M2/ Qmi iQ #2 b?Q`i HBp2/- (ke)X h?2 #`2Fi?`Qm;? +K2 i i?2 /rM Q7 i?2 kyi? +2Mim`v r?2M GB2#KMM K/2 irQ BKTQ`iMi /Bb+Qp2`B2b- (k9)- (k8)X >2 b?Qr2/ i?i irQ #bB+ T`QT2`iB2b Q7 i?2 L2riQMBM TQi2MiBH `2 HbQ biBb}2/ #v i?2 +QiM;2Mi TQi2MiBH, URV BM i?2 E2TH2` T`Q#H2K- r?B+? bim/B2b i?2 KQiBQM Q7 QM2 #Q/v `QmM/  }t2/ +2Mi`2- i?2 TQi2MiBH Bb  ?`KQMB+ 7mM+iBQM UBX2X  bQHmiBQM Q7 i?2 GTH+2 2[miBQM BM i?2 1m+HB/2M +b2- #mi Q7 i?2 GTH+2@"2Hi`KB 2[miBQM BM i?2 MQM@~i +b2Vc UkV #Qi? BM i?2 ~i M/ i?2 MQM@~i +b2b- HH #QmM/2/ Q`#Bib Q7 i?2 E2TH2` T`Q#H2K `2 +HQb2/-  T`QT2`iv /Bb+Qp2`2/ #v "2`i`M/ 7Q` i?2 L2riQMBM Hr- (j)X h?2b2 bBKBH`BiB2b #2ir22M i?2 ~i M/ i?2 +m`p2/ T`Q#H2K +QMpBM+2/ i?2 b+B2MiB}+ +QKKmMBiv i?i i?2 +QiM;2Mi TQi2MiBH rb i?2 Mim`H rv iQ 2tT`2bb ;`pBiv BM bT+2b Q7 +QMbiMi +m`pim`2X

h?2 +m`p2/ N@#Q/v T`Q#H2K #2+K2 bQK2r?i M2;H2+i2/ 7i2` i?2 #B`i? Q7 ;2M2`H `2HiBpBiv- #mi rb `2pBp2/ 7i2` i?2 /Bb+`2iBxiBQM Q7 1BMbi2BMǶb 2[miBQM b?Qr2/ i?i M N@#Q/v T`Q#H2K BM bT+2b Q7 p`B#H2 +m`pim`2 Bb iQQ +QKTHB+i2/ iQ #2 i`2i2/ rBi? MHviB+H iQQHbX AM i?2 RNNyb- i?2 _mbbBM b+?QQH Q7 +2H2biBH K2+?MB+b +QMbB/2`2/ #Qi? i?2 +m`p2/ E2TH2` M/ i?2 +m`p2/ k@#Q/v T`Q#H2Kb- (kk)- (j8)X 7i2` mM/2`biM/BM; i?i-mMHBF2 BM i?2 1m+HB/2M +b2- i?2b2 T`Q#H2Kb `2 MQi 2[mBpH2Mi- i?2 Hii2` 7BHBM; iQ #2 BMi2;`#H2- (j8)- i?2 k@#Q/v +b2 rb BMi2MbBp2Hv bim/B2/ #v b2p2`H `2b2`+?2`b Q7 i?Bb b+?QQHX JQ`2 `2+2MiHv- i?2 rQ`F Q7 .B+m- aMiQT`2i2- M/ Sû`2x@*?p2H +QMbB/2`2/ i?2 +m`p2/ N@#Q/v T`Q#H2K 7Q` N > 2 BM  M2r 7`K2rQ`F- H2/BM; iQ KMv BMi2`2biBM; `2bmHib- (8)- (3)- (d)- (Ry)- (N)- (Rj)- (R9)- (R8)- (jk)X Pi?2` `2b2`+?2`b /2p2HQT2/ i?2b2 B/2b 7m`i?2`- (Rd) (k3)- (kN)- (je)- (j3)- M/ i?2 T`Q#H2K Bb ;`QrBM; BM TQTmH`BivX

hQ 2bi#HBb? i?2 ;2QK2i`B+ Mim`2 Q7 i?2 T?vbB+H bT+2 BM i?2 Rdi? +2Mim`v- Ab+ L2riQM /2p2HQT2/ i?2 2[miBQM Q7 KQiBQM Q7 i?2 L@#Q/v T`Q#H2K BM 1m+HB/2M bT+2X h?Bb 2[miBQM Bb MQi bBKBH` iQ i?2 2[miBQMb r2 mb2 QM  /BHv #bBbX h?2 KBM TQBMi Q7 ?Bb rQ`F rb iQ mb2 i?2 +QMM2+iBQM #2ir22M ;2QK2i`v M/ /vMKB+bX AM i?2 Hi2` v2`b- T?vbB+Bbib ;`22/ iQ TT`QtBKi2 K+`Q+QbKB+ `2HBiv mbBM; +QMbiMi :mbbBM +m`pim`2X AM i?Bb rv- i?2 mM/2`biM/BM; Q7 i?2 ;2QK2i`B+ Mim`2 Q7 i?2 mMBp2`b2 #2+QK2b i?2 T`Q+2bb Q7 }M/BM; UmbBM; bi`QMQKB+H Q#b2`piBQMbV 2tBbi2M+2 Q7 i?2 Q`#Bib i?i `2 Ki?2KiB+HHv T`Qp2/ iQ 2tBbiX 1p2`vi?BM; K2MiBQM2/ #Qp2 Bb TQbbB#H2 BM i?2

j

+b2 Q7 2ti2M/BM; L2riQMǶb ;`pBiiBQMH Hr iQ k@/BK2MbBQMH bT?2`2b M/ k@/BK2MbBQMH ?vT2`#QHB+ KMB7QH/b HiQ;2i?2` rBi? `2H2pMi Ki?2KiB+H T`QQ7 i?i i?2 2tBbi2M+2 Q7 bQHmiBQMb i?i `2 bT2+B}+ iQ QMHv QM2 Q7 i?2 M2;iBp2- x2`Q- Q` TQbBiBp2 +QMbiMi :mbbBM +m`pim`2 bT+2b- #mi MQi iQ i?2 Qi?2` irQX

h?2 L2riQMBMǶb L@#Q/v T`Q#H2K 2ti2M/2/ iQ  }2H/ Q7 MQMx2`Q +QMbiMi :mbbBM +m`pim`2 Bb p2`v BKTQ`iMi 7Q` i?2 mM/2`biM/BM; ;2QK2i`v Q7 T?vbB+H bT+2X h?2 #2ii2` mM/2`biM/BM; Q7 i?2 /vMKB+b Q7 i?2 +HbbB+H 1m+HB/2M +b2 Kv +QK2 b  +QMb2[m2M+2 Q7 MHvbBb Q7 i?Bb bvbi2K r?2M i?2 +m`pim`2 i2M/b iQ x2`QX JMv #`M+?2b Q7 Ki?2KiB+b mb2  bBKBH` TT`Q+? BM T`Q#H2K bQHpBM;X

RXR amKK`v M/ Q`;MBxiBQM

h?Bb /Bbb2`iiBQM #mBH/b QM i?2 rQ`F Q7 6HQ`BM .B+m M/ 1`M2biQ Sû`2x@*?p2H QM +m`p2/ bT+2bX AM r?i 7QHHQrb- r2 rBHH Hv i?2 Ki?2KiB+H #+F;`QmM/ Q7 i?2 +m`p2/ N@#Q/v T`Q#H2K M/ /Bb+mbb BM /2iBHb i?2 Q`B;BMH `2bmHib r2 ?p2 Q#iBM2/- i?2M 2M/ i?Bb /Bbb2`iiBQM rBi?  T`QTQbH 7Q` 7mim`2 rQ`FX AM *?Ti2` k- r2 rBHH /2`Bp2 i?2 2[miBQMb Q7 KQiBQM BM +m`p2/ bT+2 Uκ ̸= 0V- M/ b?Qr ?Qr i?2v +M #2 r`Bii2M iQ BM+Hm/2 i?2 ~i +b2 Uκ = 0VX 6m`i?2`KQ`2- r2 QmiHBM2 i?2 #B7m`+iBQMb i?i Q++m` 7Q` i?2 BMi2;`Hb Q7 KQiBQM r?2M i?2 +m`pim`2 T`K2i2` Tbb2b i?`Qm;? x2`QX *?Ti2` j /2Hb rBi? i?2 #B7m`+iBQM Q7 G;`M;BM bQHmiBQMb BM i?2 BM i?2 j@M/ 9@#Q/v T`Q#H2KbX AM i?Bb +?Ti2`- r2 HbQ T`Qp2 i?`22 `2bmHibX h?2 }`bi Bb  +`Bi2`BQM 7Q` i?2 2tBbi2M+2 Q7 [m/`BHi2`H `2HiBp2 2[mBHB#`B QM i?2 2[miQ` Q7 i?2 bT?2`2X h?2 b2+QM/ b?Qrb i?i B7 irQ #Q/B2b `2 KbbH2bb M/ i?2 Qi?2` irQ `2 2[mH- i?2M b[m`2@HBF2 `2HiBp2 2[mBHB#`B 2tBbib QM bT?2`2b- #miěbm`T`BbBM;HvěMQi QM ?vT2`#QHB+ bT?2`2bX h?2 2H2K2Mi Q7 bm`T`Bb2 `Bb2b 7`QK i?2 7+i i?i- BM i?2 ;2M2`H T`Q#H2K- b[m`2@HBF2 2[mBHB#`B 2tBbi #Qi? QM i?2 ?vT2`#QHB+ bT?2`2 M/ QM i?2 bT?2`2 U2t+2Ti 7Q` i?2 +b2 r?2M i?2v `2 QM i?2 2[miQ`V- (e)X AM i?2 i?B`/ `2bmHi r2 T`Qp2 i?i B7 QMHv QM2 #Q/v Bb KbbH2bb M/ i?2 Qi?2` i?`22 `2 2[mH- bQK2 FBi2@b?T2/ `2HiBp2 2[mBHB#`B 2tBbi QM bT?2`2b- #mi MQi QM ?vT2`#QHB+ bT?2`2bX

AM *?Ti2` 9- r2 }`bi BMi`Q/m+2 i?2 `2HiBp2 2[mBHB#`B QM i?2 2[miQ`- i?2M r2 /Bb+mbb i?2 2tBbi2M+2 Q7 bQHmiBQMb 7Q` i?2 5@#Q/v T`Q#H2K- BM r?B+? 2[mBHB#`BmK bQHmiBQMb TT2` 7Q` T`iB+mH` ;2QK2i`B+H +QM};m`iBQMbX q2 HbQ b?Qr i?i i?2 `2HiBp2 2[mBHB#`B Q7 i?2 8@#Q/v T`Q#H2K rBi? 2t+iHv 7Qm` 2[mH Kbb2b Hrvb 2tBbi 7Q` i?2 bT+2b +Q``2bTQM/BM; iQ κ > 0- M/ κ < 0X JQ`2Qp2`- r2 T`Qp2 i?i i?2 b[m`2 Tv`KB/H

9

bQHmiBQMb BM S2 7Q` irQ TB`b Q7 2[mH Kbb2b /Q MQi 2tBbiX q2 +QMbB/2` i?2 THM2i`v

T`Q#H2K r?2M bQK2 #Q/B2b `2 KbbH2bbX h?2M- r2 b?Qr i?i i?2 b[m`2 Tv`KB/H bQHmiBQMb BM S2 rBi? irQ KbbH2bb #Q/B2b /Q 2tBbiX >Qr2p2`- r2 b?Qr i?i i?2 `2HiBp2

2[mBHB#`B 7Q` irQ TB`b Q7 2[mH Kbb2b /Q MQi 2tBbi BM H2X

AM *?Ti2` 8- r2 /2`Bp2  +`Bi2`BQM 7Q` i?2 2tBbi2M+2 Q7 ?2t;QMH `2HiBp2 2[mBHB#`B QM i?2 2[miQ` Q7 i?2 bT?2`2X h?2M- r2 `2+Qp2` i?2 FBi2@b?T2/ `2HiBp2 2[mBHB#`BmK 7Q` i?2 +b2 Q7 i?`22 2[mH Kbb2b M/ i?`22 KbbH2bb #Q/B2b BM +2`iBM b?T2bX

AM *?Ti2` e- r2 /2`Bp2 i?2 2[miBQMb Q7 KQiBQM 7Q` i?2 2@M/ 3@+2Mi`2 T`Q#H2Kb +Q``2bTQM/BM; iQ i?2 mTT2` ?H7 THM2 KQ/2HX h?2M- r2 bim/v i?2 /vMKB+b `QmM/ i?2 2[mBHB#`BmK TQBMi Q7 i?2 2@+2Mi2` T`Q#H2KX 6BMHHv- r2 +HbbB7v i?2 +QHHBbBQMb BM i?2 3@+2Mi2` T`Q#H2K- r?2`2b *?Ti2` d /2b+`B#2b i?2 +QM+HmbBQM M/ 7mim`2 rQ`FX

8

*?Ti2` k

1[miBQMb Q7 JQiBQM Q7 h?2 *m`p2/ L@"Q/v

S`Q#H2K

AM i?Bb +?Ti2`- r2 BMi`Q/m+2 i?2 2[miBQMb Q7 KQiBQM Q7 i?2 +m`p2/ N@#Q/v T`Q#H2K QM i?2 bm`7+2b Q7 +QMbiMi +m`pim`2X h?2M- r2 T`QpB/2 i?2 2[miBQMb Q7 KQiBQM Q7 i?2 ~i +b2 BM i?2 +QMi2ti Q7 +m`p2/ bT+2X

kXyXR h?2 TQi2MiBH

q2 BMi`Q/m+2 i?2 TQi2MiBH 7mM+iBQM QM bT+2b Q7 +QMbiMi +m`pim`2X *QMbB/2` N ≥ 2 TQBMi Kbb2b m1, . . . , mN > 0 KQpBM; QM i?2 j@bT?2`2 Q7 +QMbiMi :mbbBM +m`pim`2

κ > 0-S3 κ :={(x, y, z, w) | x2 + y2+ z2+ w2 = κ−1}, 2K#2//2/ BM R4- Q` QM i?2 ?vT2`#QHB+ j@bT?2`2 Q7 +m`pim`2 κ < 0-H3 κ :={(x, y, z, w) | x2+ y2+ z2− w2 = κ−1}, 2K#2//2/ BM i?2 JBMFQrbFB bT+2 R3,1X h?2 JBMFQrbFB bT+2 ?b 7Qm` bTiBH

+QKTQM2Mib BMbi2/ Q7 QM2 i2KTQ`H M/ i?`22 bTiBH QM2b- b mM/2`biQQ/ BM ;2M2`H `2HiBpBivX :2M2`B+HHv- r2 +M K2`;2 i?2b2 irQ KMB7QH/b BMiQ

M3

κ :={(w, x, y, z) | w2+ x2+ y2+ σz2 = κ−1, with z > 0 for κ < 0}.

h?2 MQiiBQM R3,1 2tT`2bb2b i?2 bB;Mim`2 (+, +, +, −) Q7 i?2 BMM2` T`Q/m+i- r?B+? Bb

e

G2i qi = (xi, yi, zi, wi)#2 i?2 +QQ`/BMi2b Q7 i?2 TQBMi Kbb mi M/ i?2v biBb7v i?2

+QMbi`BMi x2<sub>i</sub> + y2<sub>i</sub> + z<sub>i</sub>2+ σw2<sub>i</sub> = κ−1, r?2`2 σ Bb i?2 bB;MmK 7mM+iBQM σ := ⎧ ⎨ ⎩ +1 for κ≥ 0 −1 for κ < 0.

h?2 BMM2` T`Q/m+i #2ir22M i?2 p2+iQ`b qi = (xi, yi, zi, wi) M/ qj = (xj, yj, zj, wj) Bb

;Bp2M #v

qij = qi· qj := xixj+ yiyj + zizj+ σwiwj.

q2 /2}M2 i?2 /BbiM+2 #2ir22M i?2 #Q/B2b mi M/ mj BM S3κ- R3 M/ H3κ b

d(qi, qj) := ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ κ−12 cos−1(κqij) κ > 0 |qi− qj| κ = 0 (_−κ)−1 2 cosh−1(κqij) κ < 0

M/ i?2 ;`/B2Mi QT2`iQ` b

∇ = % ∂ ∂x, ∂ ∂y, ∂ ∂z, σ ∂ ∂w & .

q2 7m`i?2` BMi`Q/m+2 i?2 i`B;QMQK2i`B+ κ@7mM+iBQMb- r?B+? mMB7v i?2 +B`+mH` M/ ?vT2`#QHB+ i`B;QMQK2i`vX h?2 κ@bBM2 7mM+iBQMb /2}M2/ b snκ(x) := ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ κ−12 sin(κ 1 2x) κ > 0 x κ = 0 (−κ)−1 2 sinh((−κ12)x) κ < 0 i?2 κ@+QbBM2 7mM+iBQMb `2 /2}M2/ b csnκ(x) := ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ cos(κ12x) κ > 0 1 κ = 0 cosh((−κ12)x) κ < 0

d tnκ(x) := snκ(x) csnκ(x) and ctnκ(x) := csnκ(x) snκ(x)

`2bT2+iBp2HvX h?2b2 i`B;QMQK2i`B+ κ@7mM+iBQMb `2 +QMiBMmQmb rBi? `2bT2+i iQ κX G2i MQr m1, . . . , mN #2 N TQBMi Kbb2b- N ≥ 2- rBi? +Q``2bTQM/BM; TQbBiBQM p2+iQ`b q1, . . . , qN

-M/ /2MQi2 #v q = (q1, . . . , qN)i?2 +QM};m`iBQM Q7 i?2 bvbi2KX h?2M- b HQM; b κ ̸=

0-i?2 +QiM;2Mi TQi2MiBH Bb ;Bp2M #v −Uκ- r?2`2

Uκ(q) = ⎧ ⎨ ⎩ ' 1≤i<j≤Ncoth(d(mi, mj)) κ < 0 ' 1≤i<j≤Ncot(d(mi, mj)) κ > 0

Bb i?2 TQi2MiBH 7mM+iBQMX 6Q` κ = 0- TQi2MiBH Uκ(q) i2M/b iQ i?2 L2riQMBM 7Q`+2

7mM+iBQMX h?mb- i?2 TQi2MiBH Uκ(q) p`B2b +QMiBMmQmbHv rBi? i?2 +m`pim`2 κ- (e)X

ai`B;?i7Q`r`/ +QKTmiiBQMb b?Qr i?i

Uκ(q) = ( 1≤i<j≤N mimj|κ|1/2κqi· qj [(κqi· qi)(κqj· qj)− (κqi· qj)2]1/2 UkXRV 7Q` κ ̸= 0X

kXyXk 1mH2`Ƕb +HbbB+H i?2Q`2K QM ?QKQ;2M2Qmb 7mM+iBQMb

q2 rBHH bii2 1mH2`Ƕb +HbbB+H i?2Q`2K QM ?QKQ;2M2Qmb 7mM+iBQMb M/ b?Qr ?Qr Bi TTHB2b iQ i?2 +m`p2/ TQi2MiBHX

h?2Q`2K RX  7mM+iBQM F : q = (q1, ..., qn) ∈ Rm → F(q) ∈ R, r?2`2 m Bb  TQbBiBp2

BMi2;2`- Bb +HH2/ ?QKQ;2M2Qmb Q7 /2;`22 α ∈ R B7 7Q` 2p2`v η ̸= 0 M/ q ∈ Rm r2 ?p2

F (ηq) = ηα_{F (q).} h?2M

q_{· ∇F (q) = αF (q)}

aBM+2 Uκ Bb  ?QKQ;2M2Qmb 7mM+iBQM Q7 /2;`22 x2`Q- 1mH2`Ƕb i?2Q`2K

q_{· ∇F (q) = αF (q)} H2/b iQ i?2 +QM+HmbBQM i?i

q_{· ∇}qiUκ(q) = 0.

3

kXyXj .2`BpiBQM Q7 i?2 1[miBQMb Q7 JQiBQM

q2 rBHH mb2 i?2 i?2Q`v Q7 +QMbi`BM2/ G;`M;BM /vMKB+b BM Q`/2` iQ /2`Bp2 i?2 2[miBQMb Q7 KQiBQM Q7 i?2 +m`p2/ N@#Q/v T`Q#H2KX h?2 G;`M;BM Q7 i?2 +m`p2/ N@#Q/v T`Q#H2K Bb 2tT`2bb2/ b

Lκ(q, ˙q) = Tκ(q, ˙q) + Uκ(q)

r?2`2 Tκ(q, ˙q) = 1<sub>2</sub>'N<sub>i=1</sub>mi( ˙q⊙ ˙q) Bb i?2 FBM2iB+ 2M2`;v- r?2`2 ⊙ Bb i?2 BMM2` T`Q/m+iX

h?2M- ++Q`/BM; iQ i?2 i?2Q`v Q7 +QMbi`BM2/ G;`M;BM /vMKB+b- i?2 2[miBQMb 2[miBQM Q7 KQiBQM ?b i?2 7Q`K

d dt % ∂Lκ ∂ ˙qi & − ∂L_∂qκ i − λ i κ(t) ∂fi κ ∂qi = 0, i = 1, ..., N, r?2`2 fi

κ = (qi⊙ qi)− κ−1 Bb i?2 7mM+iBQM i?i +?`+i2`Bx2b i?2 +QMbi`BMib fκi = 0, i =

1, 2, ..., N- M/ λi

κ `2 i?2 G;`M;2 KmHiBTHB2`b ;Bp2M #v λiκ =−κmi( ˙q⊙ ˙q), i = 1, ...., N.

h?2M- i?2 2[miBQM Q7 KQiBQM `2 ;Bp2M #v i?2 bvbi2K Q7 /Bz2`2MiBH 2[miBQMb

miq¨i = ∇qiUκ(q) − miκ( ˙qi · ˙qi)qi, qi · qi = κ −1_{, q} i · ˙qi = 0, i = 1, . . . , N . r?2`2 κ ̸= 0 M/ ∇qiUκ(q) = N ( j=1 j̸=i mimj|κ|3/2(κqj· qj)[(κqi· qi)qj− (κqi· qj)qi] [σ(κqi · qi)(κqj· qj)− σ(κqi· qj)2]3/2 . UkXkV

hQ F22T i?2 #Q/B2b QM i?2 KMB7QH/b- r2 ?p2 iQ bbmK2 i?i- i bQK2 BMBiBH iBK2- i?2 TQbBiBQM p2+iQ`b M/ i?2 p2HQ+Biv p2+iQ`b biBb7v i?2 2N +QMbi`BMib

κqi· qi = 1, qi· ˙qi = 0.

lbBM; i?2b2 +QMbi`BMib r2 +M r`Bi2 i?i

∇qiUκ(q) = N ( j=1,j̸=i mimj|κ|3/2[qj − (κqi· qj)qi] ) σ− σ (κqi· qj)2 *3/2 , κ̸= 0, i = 1, ..., N.

N

kXR 1ti2MbBQM iQ i?2 ~i +b2

h?2 BM+QMp2MB2M+2 Q7 ?pBM; iQ mb2 irQ /Bz2`2Mi b2ib Q7 2[miBQMb- QM2 7Q` κ ̸= 0 M/ MQi?2` 7Q` κ = 0- B7 r2 rMi iQ +QMbB/2` i?2 T`Q#H2K 7Q` Mv κ ∈ R- ?b #22M Qp2`+QK2 #v 6X .B+m BM (Ry)X >2 }`bi MQiB+2/ i?i

2qij = q_i2+ q_j2− q2 ij, r?2`2 qij := ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ [(xi− xj)2+ (yi− yj)2+ (zi− zj)2+ (wi− wj)2]1/2 κ > 0 [(xi− xj)2+ (yi− yj)2+ (zi− zj)2]1/2 κ = 0 [(xi− xj)2+ (yi− yj)2+ (zi− zj)2− (wi− wj)2]1/2 κ < 0

Bb i?2 1m+HB/2M /BbiM+2 #2ir22M mi M/ mj BM R4 7Q` κ > 0- #mi BM R3 7Q` κ = 0- M/

i?2 +Q``2bTQM/BM; /BbiM+2 BM i?2 JBMFQrbFB bT+2 R3,1 7Q` κ < 0X HH Q7 i?2K iF2M

#2ir22M i?2 TQBMi Kbb2b mi M/ mjX LQiB+2 i?i i?2 JBMFQrbFB /BbiM+2 /Q2b MQi

biBb7v i?2 i`BM;H2 BM2[mHBiv qik ≤ qij + qik- #mi Bb Hrvb MQM@M2;iBp2X

q`BiBM; i?2 TQi2MiBH 7mM+iBQM Uκ mbBM; i?2 #Qp2 `2HiBQMb?BT r2 Q#iBM  M2r

2tT`2bbBQM Q7 i?2 7Q`+2 7mM+iBQM-Vκ(q) = ( 1≤i<j≤N mimj(κq2i + κqj2− κq2ij) [2(κq2 i + κqj2)q2ij− κ(qi2− q2j)2− κqij4]1/2 ,

r?B+? +M #2 Tmi BMiQ i?2 7Q`K

Vκ(q) = ( 1≤i<j≤N mimj + 1₋ κq2ij 2 , qij + 1− κq2ij 4 ,1/2. UkXjV

LQiB+2 i?i i?2 7Q`KmH Q7 Uκ BM UkXRV +MMQi #2 2ti2M/2/ iQ i?2 ~i +b2- #mi i?2 `B;?i

?M/ bB/2 Q7 UkXjV KF2b b2Mb2 7Q` κ = 0X h?2M 7Q` κ = 0- r2 `2+Qp2` i?2 +HbbB+H L2riQMBM 7Q`+2 7mM+iBQM V0(q) = ( 1≤i<j≤N mimj qij .

.2}MBiBQM RX *QMbB/2`  7KBHv Q7 ?QKQ;2M2Qmb 7mM+iBQMb Fκ: Rm → R, κ ∈ R- i?i Bb

Ry

pHm2 Q7 α /Q2b MQi /2T2M/ QM κ- r?2`2 α Bb i?2 /2;`22 Q7 FκX A7 i?2 pHm2 Q7 α /2T2M/b

QM κ- i?2M i?2 7KBHv 2tT2`B2M+2b  +?M;2 BM ?QKQ;2M2Biv i i?2 TQBMib κ r?2`2 i?2 pHm2 Q7 α +?M;2bX

S`QTQbBiBQM RX h?2 7Q`+2 7mM+iBQM Vκ ;Bp2M BM UkXjV Bb +QMiBMmQmb BM κ 7Q` HH κ ∈ R

M/ i?2 7KBHv ?b  +?M;2 BM ?QKQ;2M2Biv i κ = 0- MK2Hv Bi T`QpB/2b ?QKQ;2M2Qmb 7mM+iBQMb Q7 /2;`22 0 7Q` κ ̸= 0- #mi  ?QKQ;2M2Qmb 7mM+iBQM Q7 /2;`22 −1 i κ = 0X S`QQ7X h?2 +QMiBMmBiv Q7 Vκ BM κ Bb Q#pBQmb 7`QK UkXjVX 6Q` κ ̸= 0- i?2 7Q`+2 7mM+iBQM Uκ

BM UkXRV Bb i?2 bK2 b Vκ BM UkXjVX lbBM; i?2 2tT`2bbBQM Q7 ∇qiUκ(q) BM UkXkV- Bi 7QHHQrb

i?i q_{· ∇}qiUκ(q) = N ( j=1 j̸=i mimj|κ|3/2(κqj · qj)[(κqi· qi)(qj · qi)− (κqi· qj)(qi· qi)] [σ(κqi· qi)(κqj· qj)− σ(κqi· qj)2]3/2 = 0

bBM+2 i?2 #`+F2ib pMBb? BM HH i?2 MmK2`iQ`bX q?2M κ ̸= 0- i?2 +m`p2/ 7Q`+2 7mM+iBQM Uκ 2tT2`B2M+2b MQ +?M;2 BM ?QKQ;2M2BivX q2 +QM+Hm/2 7`QK UkXRV i?i i?2 /2;`22 Q7

?QKQ;2M2Biv Bb yX 6Q` κ = 0- r2 Q#iBM i?i

q· ∇U0(q) =−U0(q).

h?2`27Q`2- ++Q`/BM; iQ 1mH2`Ƕb 7Q`KmH- i?2 /2;`22 Q7 ?QKQ;2M2Biv Q7 U0 Bb −1-  7+i

i?i +QKTH2i2b i?2 T`QQ7X

h?2 2[miBQMb Q7 KQiBQM iF2 i?2 7Q`K

¨ qi = N ( j=1,j̸=i mj -qj − + 1₋ κq2ij 2 , qi . + 1− κq2ij 4 ,3/2 q3 ij − κ( ˙qi· ˙qi)qi, i = 1, . . . , N.

6m`i?2` AMi`Q/m+BM; i?2 i`Mb7Q`KiBQMb

ωi = wi− |κ|−1/2, i = 1, . . . , N,

r?B+? b?B7i i?2 Q`B;BM Q7 i?2 +QQ`/BMi2 bvbi2K iQ i?2 LQ`i? SQH2 Q7 i?2 bT?2`2b M/ KFBM; i?2 MQiiBQMb

RR

i?2 2[miBQMb Q7 KQiBQM iF2 i?2 7Q`K

¨ri = N ( j=1,j̸=i mj -rj − + 1₋ κr2ij 2 , ri+ Rr2 ij 2 . + 1− κr2ij 4 ,3/2 r3 ij − (˙ri· ˙ri)(κri+ R), i = 1, . . . , N, M/ i?2 2N +QMbi`BMib #2+QK2 κr2_i + 2|κ|1/2_ω i = 0, κri· ˙ri+|κ|1/2ω˙i = 0

PM +QKTQM2Mib- i?2b2 2[miBQMb +M #2 r`Bii2M b ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ¨ xi ='Nj=1,j̸=i mj -xj− + 1−κr2₂ij , xi . + 1−κr24ij ,3/2 r3 ij − κ(˙ri· ˙ri)xi ¨ yi ='N_j=1,j̸=i mj -yj− + 1−κr22ij , yi . + 1−κr24ij ,3/2 r3 ij − κ(˙ri· ˙ri)yi ¨ zi ='N_j=1,j̸=i mj -zj− + 1−κr22ij , zi . + 1−κr2₄ij ,3/2 r3 ij − κ(˙ri· ˙ri)zi ¨ ωi ='Nj=1,j̸=i mj -ωj− + 1−κr22ij , ωi . + 1−κr24ij ,3/2 r3 ij − [κωi + (σκ]1/2)(˙ri· ˙ri)ωi

6Q` κ = 0 r2 `2+Qp2` i?2 +HbbB+H L2riQMBM 2[miBQMb Q7 i?2 N@#Q/v

T`Q#H2K-¨ ri = N ( j=1,j̸=i mj(rj − ri) r3 ij , i = 1, . . . , N.

q?2M κ = 0- i?2 TQbBiBQM p2+iQ`b ri = (xi, yi, zi, 0) `2 7`22 Q7 +QMbi`BMib M/ i?2

p2HQ+BiB2b ˙ri = ( ˙xi, ˙yi, ˙zi, 0)- r?B+? Bb HBF2 i?BMFBM; i?i i?2 KQiBQM BM R3 iF2b TH+2 BM 

?vT2`THM2 Q7 R4X

kXk "B7m`+iBQM Q7 i?2 BMi2;`Hb Q7 KQiBQM

h?2 BMi2;`Hb Q7 KQiBQM `2 7mM+iBQMb Q7 i?2 T?b2@bT+2 +QQ`/BMi2b i?i `2 +QMbiMi HQM; Q`#BibX G2i mb BMp2biB;i2 i?2 BMi2;`Hb Q7 KQiBQM 7Q` i?2 #Qp2 bvbi2K Q7 2[miBQMbX

Rk

kXkXR h?2 BMi2;`Hb Q7 i?2 +2Mi`2 Q7 Kbb M/ i?2 HBM2` KQK2MimK

amKKBM; mT mir¨i BM i?2 2[miBQMb Q7 KQiBQM 7`QK i = 1 iQ i = N r2 Q#iBM i?2 B/2MiBiv

N ( i=1 mi¨ri = N ( i=1 N ( j=1,j̸=i mjmi r2 ij 2 + κri+ R , + 1− κr2ij 4 ,3/2 r3 ij − N ( i=1 mi(˙ri· ˙ri)(κri+ R). 6Q` κ = 0- r2 ?p2 'N

i=1mi¨ri = 0X AMi2;`iBM; i?Bb B/2MiBiv r2 Q#iBM i?2 i?`22 BMi2;`Hb

Q7 i?2 HBM2` KQK2MimK BM i?2 ~i

+b2-N

(

i=1

mi˙ri = a

r?2`2 a = (a1, a2, a3)Bb  +QMbiMi p2+iQ`X AMi2;`iBM; ;BM- r2 Q#iBM N

(

i=1

miri = at + b

r?2`2 b = (b1, b2, b3)Bb MQi?2` +QMbiMi p2+iQ`X

"v iFBM; i?2 Q`B;BM Q7 i?2 +QQ`/BMi2 bvbi2K i i?2 +2Mi`2 Q7 Kbb- r2 Q#iBM

N ( i=1 mi˙ri = 0 and N ( i=1 miri = 0,

r?B+? K2Mb i?2 +2Mi`2 Q7 Kbb Bb }t2/ `2HiBp2 iQ i?2 +QQ`/BMi2 bvbi2KX >2M+2- i?2 7Q`+2b +iBM; QM i?2 +2Mi`2 Q7 Kbb +M+2H 2+? Qi?2`X

P#pBQmbHv- i?2b2 BMi2;`Hb /Q MQi b?Qr mT r?2M κ ̸= 0- i?mb r2 2M+QmMi2`  #B7m`+iBQM i κ = 0X *H2`Hv- i?Bb #B7m`+iBQM K2Mb  +?M;2 BM i?2 T?b2@bT+2 TQ`i`BiX

kXkXk h?2 BMi2;`H Q7 2M2`;v

hFBM; miq¨i· ˙qi M/ bmKKBM; mT 7`QK i = 1 iQ i = N- r2 ?p2 N ( i=1 miq¨i· ˙qi = N ( i=1 ˙qi· ∇qiVκ(q)− N ( i=1 mi( ˙qi· ˙qi)(κqi· ˙qi) = d dtVκ(q).

Rj

"v BMi2;`iBQM r2 Q#iBM i?2 2M2`;v BMi2;`H

Hκ(q, ˙q) = Tκ(q, ˙q)− Vκ(q) = h,

r?2`2 h Bb  +QMbiMi- Hκ Bb i?2 >KBHiQMBM 7mM+iBQM- M/ Tκ Bb i?2 FBM2iB+ 2M2`;vX

7i2` TTHvBM; i?2 i`Mb7Q`KiBQMb ωi = wi− |κ|−1/2- i?2 FBM2iB+ 2M2`;v #2+QK2b

Tκ(r, ˙r) = 1 2 N ( i=1 mi[κr2i + 2(σκ)1/2ωi+ 1]( ˙ri· ˙ri).

h?2M i?2 BMi2;`H Q7 2M2`;v 7Q` i?2 bvbi2K iF2b i?2 7Q`K

1 2 N ( i=1 mi[κri2+ 2(σκ)1/2ωi+ 1](˙ri· ˙ri)− ( 1≤i<j≤N mimj + 1₋ κr2ij 2 , rij + 1− κr2ij 4 ,1/2 = h.

6Q` κ = 0- r2 Q#iBM i?2 BMi2;`H Q7 i?2 L2riQMBM 2[miBQMb,

1 2 N ( i=1 mi( ˙x2i + ˙yi2+ ˙zi2)− ( 1≤i<j≤N mimj rij = h. h?mb MQ #B7m`+iBQMb Q++m` BM i?Bb +b2X

kXkXj h?2 BMi2;`Hb Q7 i?2 iQiH M;mH` KQK2MimK

q2 /2}M2 i?2 iQiH M;mH` KQK2MimK b

N

(

i=1

miqi∧ ˙qi

r?2`2 ∧ Bb i?2 2ti2`BQ` Ur2/;2V T`Q/m+i BM i?2 :`bbKM H;2#` Qp2` R4_, (Ry)X h?2

iQiH M;mH` KQK2MimK K2bm`2b i?2 `QiiBQM Q7 i?2 bvbi2K `2HiBp2 iQ i?2 bBt THM2b ;Bp2M #v 2p2`v irQ Q7 i?2 7Qm` t2b i?i 7Q`K i?2 +QQ`/BMi2 bvbi2K Q7 R4X h?Bb [mMiBiv

Bb +QMb2`p2/ 7Q` i?2 2[miBQMb Q7 KQiBQM b b?QrM BM

(Ry)-N

(

i=1

R9

r?2`2

c = cwxew∧ ex+ cwyew ∧ ey+ cwzew∧ ez+ cxyex∧ ey+ cxzex∧ ez+ cyzey ∧ ez,

rBi? cwx, cwy, cwz, cxy, cxz, cyz ∈ R, M/ i?2 p2+iQ`b Q7 i?2 biM/`/ #bBb Q7 R4 #2BM;

ex = (1, 0, 0, 0), ey = (0, 1, 0, 0), ez = (0, 0, 1, 0), ew = (0, 0, 0, 1). PM +QKTQM2Mib- r2 ?p2 bBt b+H` BMi2;`Hb-N ( i=1 mi(xiy˙i− ˙xiyi) = cxy, N ( i=1 mi(xiz˙i− ˙xizi) = cxz, N ( i=1 mi(yiz˙i− ˙yizi) = cyz, N ( i=1 mi(wix˙i− ˙wixi) = cwx, N ( i=1 mi(wiy˙i− ˙wiyi) = cwy, N ( i=1 mi(wiz˙i− ˙wizi) = cwz.

lbBM; i?2 i`Mb7Q`KiBQM ωi = wi− |κ|−1/2- i?2b2 BMi2;`Hb iF2 i?2 7Q`K N ( i=1 mi(xiy˙i− ˙xiyi) = cxy, N ( i=1 mi(xiz˙i− ˙xizi) = cxz, N ( i=1 mi(yiz˙i− ˙yizi) = cyz, N ( i=1 mi˙xi+ (σκ)1/2 N ( i=1 mi(wix˙i− ˙wixi) = (σκ)1/2cwx, N ( i=1 mi˙yi+ (σκ)1/2 N ( i=1 mi(wiy˙i− ˙wiyi) = (σκ)1/2cwy, N ( i=1 mi˙zi+ (σκ)1/2 N ( i=1 mi(wiz˙i− ˙wizi) = (σκ)1/2cwz.

LQiB+2 i?i- 7Q` κ = 0- i?2 Hbi i?`22 BMi2;`Hb #2+QK2 HBM2` KQK2MimK BMi2;`Hb- bQ M BMi2`2biBM; FBM/ Q7 #B7m`+iBQM Q++m`bX 6`QK i?2 T?vbB+H TQBMi Q7 pB2r- i?Bb TT2`b Mim`H- ;Bp2M i?i  dz+m`p2/Ǵ /BK2MbBQM- r?B+? +QMp2vb i?2 +QMb2`piBQM Q7  `QiiBQMH +QKTQM2Mi Q7 i?2 KQiBQM- dzbi`B;?i2Mb mTǴ r?2M κ ;Q2b 7`QK MQMx2`Q pHm2b iQ x2`Q-M/ i?mb ;2ib i`Mb7Q`K2/ BMiQ  HBM2` +QKTQM2Mi Q7 i?2 KQiBQMX

R8

*?Ti2` j

"B7m`+iBQM Q7 h?2 G;`M;BM P`#Bib

h?Bb +?Ti2` ?b #22M Tm#HBb?2/ BM _1GhAo1 1ZlAGA"_A AL *l_o1. _1ah_A*h1. 9@"P.u S_P"G1Ja BM h?2 *M/BM Ji?2KiB+H "mHH2iBM (k)X

AM i?Bb +?Ti2` r2 +QMbB/2` i?2 #B7m`+iBQM Q7 bQHmiBQMb Q7 i?2 2[miBQMb Q7 KQiBQM QM bT?2`2b- S2

κ M/ ?vT2`#QHB+ bT?2`2b H2κX q2 /2H rBi? i?2 G;`M;BM Q`#Bib- r?B+? `2

2[mBHi2`H i`BM;H2b BM i?2 j@#Q/v T`Q#H2KX q2 +QMbB/2` i?2 KQiBQM Q7 7Qm` #Q/B2b QM k@/BK2MbBQMH bm`7+2b Q7 +QMbiMi +m`pim`2 κX 6Q` κ > 0 r2 mb2 b  KQ/2H i?2 bT?2`2b Q7 `/Bmb 1/√κX h?Bb bT?2`2 Bb 2K#2//2/ BM R3 rBi? i?2 1m+HB/2M K2i`B+- r2 /2MQi2

Bi #v S2

κX 6Q` κ = 0 r2 iF2 i?2 1m+HB/2M THM2 R2- M/ 7Q` κ < 0 r2 iF2 i?2 mTT2`

T`i Q7 i?2 ?vT2`#QHQB/

x2_{+ y}2_{− z}2 _=−1/√_−κ, UjXRV

2K#2//2/ BM i?2 JBMFQrbFB bT+2 R2,1- i?i Bb R3 2M/Qr2/ rBi? i?2 GQ`2Mx BMM2`

T`Q/m+i U7Q` a, b ∈ R3_{, a⊙ b = a}

xbx+ ayby− azbzVX h?Bb bT+2 Bb FMQrM b i?2 ?vT2`#QHB+

bT?2`2 Q` i?2 Tb2m/Q bT?2`2- M/ Bi Bb /2MQi2/ #v H2 κX

LQr- r2 rBHH ``M;2 i?2b2 Q#D2+ib BM R3- KBMiBMBM; i?2 /Bz2`2Mi K2i`B+ 7Q` i?2

bT?2`2 M/ i?2 Tb2m/Q bT?2`2- bm+? i?i i?2v HH ?p2  +QKKQM TQBMi i r?B+? HB2 HH i?2 MQ`i? TQH2b Q7 i?2 bT?2`2b M/ i?2 p2`iB+2b Q7 i?2 ?vT2`#QHB+ bT?2`2b- iQ HH Q7 r?B+? i?2 THM2 R2 Bb iM;2MiX q2 }t i?2 Q`B;BM Q7 i?2 M2r +QQ`/BMi2 bvbi2K i i?Bb TQBMiX

AM Qi?2` rQ`/b- r2 i`MbHi2 i?2 Q`B;BM iQ i?2 MQ`i? TQH2 Q7 i?2 bT?2`2 M/ i?2 Tb2m/Q bT?2`2- #mbBM; MQiiBQM r2 F22T i?2 bK2 MQiiBQM 7Q` i?2b2 Q#D2+ib- i?2M r2 +M r`Bi2

S2 κ :={(x, y, z) | κ(x2+ y2+ z2) + 2κ 1 2z = 0} for κ > 0, H2 κ :={(x, y, z) | κ(x2+ y2− z2) + 2|κ| 1 2z = 0, z ≥ 0} for κ < 0.

Re

*QMbB/2` MQr 7Qm` TQBMi Kbb2b- mi > 0, i = 1, 2, 3, 4- r?Qb2 TQbBiBQM p2+iQ`b-

p2HQ+BiB2b-M/ ++2H2`iBQMb `2 ;Bp2M #v

ri = (xi, yi, zi), ˙ri = ( ˙xi, ˙yi, ˙zi), ¨ri = (¨xi, ¨yi, ¨zi), i = 1, 2, 3, 4.

h?2M- b b?QrM BM (N)- i?2 2[miBQMb Q7 KQiBQM iF2 i?2 7Q`K ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ¨ xi ='N<sub>j=1,j̸=i</sub> mj -xj− + 1−κr22ij , xi . + 1−κr24ij ,3/2 r3 ij − κ(˙ri· ˙ri)xi ¨ yi ='N<sub>j=1,j̸=i</sub> mj -yj− + 1−κr22ij , yi . + 1−κr2<sub>4</sub>ij ,3/2 r3 ij − κ(˙ri· ˙ri)yi ¨ zi ='Nj=1,j̸=i mj -zj− + 1−κr22ij , zi . + 1−κr24ij ,3/2 r3 ij − (˙ri · ˙ri)(κzi+ σ|κ|1/2), i = 1, 2, 3, 4, UjXkV r?2`2 σ = 1 7Q` κ ≥ 0- σ = −1 7Q` κ < 0- M/ rij := ⎧ ⎨ ⎩ [(xi− xj)2+ (yi− yj)2+ (zi− zj)2]1/2 for κ≥ 0 [(xi− xj)2+ (yi− yj)2− (zi− zj)2]1/2 for κ < 0

7Q` i, j ∈ {1, 2, 3, 4}X h?2 #Qp2 bvbi2K ?b 2B;?i +QMbi`BMib- MK2Hv

κ(x2₁+ y_i2+ σz_i2) + 2_|κ|1/2zi = 0,

κri· ˙ri+|κ|1/2˙zi = 0, i = 1, 2, 3, 4.

A7 biBb}2/ i M BMBiBH BMbiMi- i?2b2 +QMbi`BMib `2 biBb}2/ 7Q` HH iBK2 #2+mb2 i?2 b2ib S2

κ,R2- M/ H2κ `2 BMp`BMi 7Q` i?2 2[miBQMb Q7 KQiBQM- (e)X LQiB+2 i?i 7Q` κ = 0

r2 `2+Qp2` i?2 +HbbB+H L2riQMBM 2[miBQMb Q7 i?2 4@#Q/v T`Q#H2K QM i?2 1m+HB/2M THM2- MK2Hv ¨ri = N ( j=1,j̸=i mj(rj − ri) r3 ij , r?2`2 ri = (xi, yi, 0), i = 1, 2, 3, 4.

Rd

jXR _2HiBp2 1[mBHB#`B

Ai Bb r2HH FMQrM i?i BM i?2 +m`p2/ N@#Q/v T`Q#H2K- i?2 iQiH 2M2`;v M/ i?2 M;mH` KQK2MimK `2 }`bi BMi2;`Hb- #mi i?2 HBM2` KQK2MimK Bb MQ HQM;2`  +QMbiMi Q7 KQiBQM- r?B+? Bb  #B; /Bz2`2M+2 rBi? i?2 1m+HB/2M +b2 (RR)X h?2 ;QH Q7 i?Bb b2+iBQM Bb iQ /2b+`B#2 T`iB+mH` bQHmiBQMb 7Q` i?Bb T`Q#H2K- i?2 bBKTH2bi QM2b- +HH2/ `2HiBp2 2[mBHB#`BX h?2 7Q`KH /2}MBiBQM Bb i?2 7QHHQrBM;X

.2}MBiBQM kX _2HiBp2 2[mBHB#`B `2 bQHmiBQMb Q7 i?2 +m`p2/ N@#Q/v T`Q#H2K BM r?B+? i?2 KmimH /BbiM+2b KQM; i?2 T`iB+H2b `2KBM +QMbiMi 7Q` HH iBK2 t ∈ RX h?i Bb i?2 T`iB+H2b KQp2 HBF2  `B;B/ #Q/vX

aQ BM Q`/2` iQ bim/v `2HiBp2 2[mBHB#`B r2 Kmbi MHvx2 HH BbQK2i`B2b 7Q` #Qi? i?2 bT?2`2 S2

κ M/ i?2 Tb2m/Q@bT?2`2 H2κX ++Q`/BM; iQ i?2 #Qp2 /2}MBiBQM- i?2 `2HiBp2

2[mBHB#`B rBHH #2 i?2 bQHmiBQMb Q7 i?2 2[miBQMb Q7 KQiBQM r?B+? `2 BMp`BMi mM/2` i?2 +iBQM Q7 i?2 BbQK2i`v ;`QmTb 7Q` i?2 `2bT2+iBp2 bm`7+2b Q7 TQbBiBp2 M/ M2;iBp2 +m`pim`2X

jXRXR _2HiBp2 2[mBHB#`B 7Q` TQbBiBp2 κ

h?Bb Bb i?2 bBKTH2bi +b2- #2+mb2 r2 FMQr i?i HH BbQK2i`B2b BM R3 `2 `QiiBQMbc M/

h?2 S`BM+BTH tBb h?2Q`2K bii2b i?i Mv `QiiBQM BM R3- Bb `QmM/  }t2/ tBb (Re)X

aQ BM i?Bb +b2- rBi?Qmi HQbb Q7 ;2M2`HBiv r2 +M bbmK2 i?i i?2 `QiiBQM Bb `QmM/ i?2 z@tBb- M/ r2 ?p2 i?i-  `2HiBp2 2[mBHB#`BmK Bb  bQHmiBQM Q7 i?2 2[miBQMb Q7 KQiBQM i?i Bb BMp`BMi mM/2` i?2 +iBQM Q7 i?2 BbQK2i`v ;Bp2M #v  `QiiBQM Ki`Bt

A(t) = ⎛ ⎜ ⎝ cos t − sin t 0 sin t cos t 0 0 0 1 ⎞ ⎟ ⎠ , UjXjV

jXRXk _2HiBp2 2[mBHB#`B 7Q` M2;iBp2 κ

q2 }`bi +QMbB/2` i?2 k@/BK2MiBQMH q2B2`bi`bb KQ/2H- r?B+? Bb #mBHi QM QM2 Q7 i?2 b?22ib Q7 ?vT2`#QHQB/ Q7 irQ b?22ib `2T`2b2Mi2/ #v UjXRV- BM i?2 JBMFQrbFB bT+2 (R2,1<sub>,⊙) 7Q`</sub>

κ < 0X h?2 HQ`2Mix BMM2` T`Q/m+i #2ir22M i?2 p2+iQ`b qi = (xi, yi, zi)M/ qj = (xj, yj, zj)

Bb ;Bp2M #v

R3

h?Bb bm`7+2 +M #2 `2T`2b2Mi2/ #v i?2 mTT2` b?22i Q7 i?2 ?vT2`#QHB+ bT?2`2 rBi? z > 0X 1[mBpH2MiHv- i?Bb bm`7+2 #2BM; b  Tb2m/Qb?2`2 Q7 BK;BM`v `/Bmb (iR)- bm+? i?i (iR)2 _{= κ}−1X

 HBM2` i`Mb7Q`KiBQM T : R2,1 <sub>→ R</sub>2,1 Bb Q`i?Q;QMH B7 T (x) ⊙ T(x) = x ⊙ x 7Q`

x _{∈ R}2,1X h?2 b2i Q7 i?2b2 i`Mb7Q`KiBQMb rBi? i?2 GQ`2Mix BMM2` T`Q/m+i 7Q`K i?2

Q`i?Q;QMH ;`QmT O(R2,1_{) = {T (x) ⊙ T (x) = x ⊙ x, x ∈ R}2,1_{}- ;Bp2M #v Ki`B+2b rBi?}

/2i2`KBMMi ±1X

h?2 bm#;`QmT Q7 SO(R2,1<sub>) ={T ∈ O(R</sub>2,1<sub>), det(T ) = 1} Bb +HH2/ i?2 bT2+BH Q`i?Q;QMH</sub>

;`QmT rBi? /2i2`KBMMi +1X h?2 bm#b2i G(R2,1₎ Bb MQi?2` bm#;`QmT Q7 O(R2,1₎ r?B+?

Bb 7Q`K2/ #v i?2 i`Mb7Q`KiBQM T i?i H2p2b H2

κ BMp`BMiX JQ`2Qp2`- G(R2,1)?b i?2

+HQb2/ GQ`2Mix bm#;`QmT- Lor(R2,1_{) := G(R}2,1_{)∩ SO(R}2,1_).

G2i Lor(H2

κ,⊙) #2 i?2 ;`QmT Q7 HH Q`i?Q;QMH i`Mb7Q`KiBQMb Q7 /2i2`KBMMi R

i?i KBMiBM i?2 mTT2` T`i Q7 i?2 ?vT2`#QHQB/ BMp`BMi Ui?2 GQ`2Mix ;`QmT 7Q`K2/ #v HH BbQK2i`B2b Q7 H2

κV Ub22 (RR- Rj- jk) 7Q` KQ`2 /2iBHbVX TTHvBM; i?2 +Q``2bTQM/BM;

S`BM+BTH tBb h?2Q`2K (jy) iQ Lor(H2

κ,⊙)- r?B+? bii2b i?i Mv 1−T`K2i2` bm#;`QmT

Q7 Lor(L2<sub>,⊙) +M #2 r`Bii2M- BM  T`QT2` #bBb- b</sub> A(t) = P ⎛ ⎜ ⎝ cos t <sub>− sin t 0</sub> sin t cos t 0 0 0 1 ⎞ ⎟ ⎠ P−1, Q` A(t) = P ⎛ ⎜ ⎝ 1 0 0 0 cosh t sinh t 0 sinh t cosh t ⎞ ⎟ ⎠ P−1, Q` A(t) = P ⎛ ⎜ ⎝ 1 <sub>−t</sub> t t 1<sub>− t</sub>2<sub>/2 t</sub>2<sub>/2</sub> t <sub>−t</sub>2<sub>/2 1 + t</sub>2<sub>/2</sub> ⎞ ⎟ ⎠ P−1, r?2`2 P ∈ Lor(H2

κ,⊙)X h?2M- Mv BbQK2i`v Q7 Lor(H2κ,⊙) +M #2 r`Bii2M b 

+QKTQbBiBQM Q7 bQK2 Q7 i?2 #Qp2 i?`22 i`Mb7Q`KiBQMb- +HH2/ `2bT2+iBp2Hv- 2HHBTiB+-?vT2`#QHB+ M/ T`#QHB+ i`Mb7Q`KiBQMbX aQ BM i?Bb +b2- i?2 `2HiBp2 2[mBHB#`B QM i?2

RN

Tb2m/Q bT?2`2 `2 i?2 bQHmiBQMb Q7 i?2 2[miBQMb Q7 KQiBQM r?B+? `2 BMp`BMi mM/2` HH BbQK2i`v Q7 Lor(H2

κ,⊙)X

jXk h?2 +b2 Q7 TQbBiBp2 +m`pim`2, S

2 κ

"v i?2 T`2pBQmb /Bb+mbbBQM- BM i?Bb +b2 r2 ?p2 iQ }M/ i?2 BMBiBH +QM/BiBQMb r?B+? H2/ iQ bQHmiBQMb BMp`BMi mM/2` i?2 +iBQM Q7 i?2 BbQK2i`v ;Bp2M #v i?2 `QiiBQM Ki`Bt /2}M2/ #v 2[miBQM UjXjVX

6B`bi- r2 BMi`Q/m+2 bT?2`B+H +QQ`/BMi2b (ϕ, ω)- r?B+? r2`2 Q`B;BMHHv mb2/ BM (N) 7Q` i?2 +b2 N = 3- iQ /2i2+i `2HiBp2 2[mBHB#`B QM M/ M2` i?2 2[miQ` Q7 S2

κ- r?2`2 ϕ

K2bm`2b i?2 M;H2 7`QK i?2 x@tBb BM i?2 xy@THM2- r?BH2 ω Bb i?2 ?2B;?i QM i?2 p2`iB+H z@tBbX AM i?2b2 M2r +QQ`/BMi2b- i?2 3 +QMbi`BMib 7Q` i?2 Q`B;BMH 2[miBQMb Q7 KQiBQM UjXkV #2+QK2

x2_i + y2_i + ω2_i + 2κ−1/2ωi = 0, i = 1, 2, 3, 4, 5, ..N. UjX9V

qBi? i?2

MQiiBQM-Ωi = x2i + yi2 =−κ−1/2ωi(κ1/2ωi+ 2)≥ 0, ωi ∈ [−2κ−1/2, 0], i = 1, 2, 3, 4, ..N

r?2`2 2[mHBiv Q++m`b r?2M i?2 #Q/v Bb i i?2 LQ`i? Q` i?2 aQmi? SQH2 Q7 i?2 bT?2`2-i?2 (ϕ, ω)@+QQ`/BMi2b `2 ;Bp2M #v bT?2`2-i?2 i`Mb7Q`KiBQMb

xi = Ω1/2i cos ϕi, yi = Ω1/2i sin ϕi.

h?mb i?2 2[miBQMb Q7 KQiBQM UjXkV iF2 i?2 7Q`K

¨ ϕi = Ω−1/2i N ( j=1,j̸=i mjΩ1/2j sin(ϕj − ϕi) ρ3 ij(1− κρ2 ij 4 )3/2 − ϕ˙iΩ˙i Ωi , ¨ ωi = Ω−1/2i N ( j=1,j̸=i mj -ωj + ωi+ κρ2 ij 2 (ωi+ κ−1/2) . ρ3 ij + 1₋κρ2ij 4 ,3/2 UjX8V − (κωi+ κ1/2)( ˙ Ω2 i 4Ωi + ˙ϕi2Ωi+ ˙ωi2), r?2`2 ˙Ωi =−2κ−1/2ω˙i(κ1/2ωi+ 1)

ky

ρ2_ij = Ωi+ Ωj− 2Ω1/2i Ω 1/2

j cos(ϕi− ϕj) + (ωi− ωj)2, i, j = 1, 2, 3, 4, .., N i̸= j.

jXkXR _2HiBp2 2[mBHB#`B QM i?2 2[miQ` 7Q` N = 3

"v }`bi b22FBM; `2HiBp2 2[mBHB#`B QM i?2 2[miQ`- r2 iF2 ω = −κ−1/2- bQ r2 ?p2

ωi =−κ−1/2, ˙ωi = 0, Ωi = κ−1, Ω˙i = 0,

ρ2_ij =_−2κ−1[1_{− cos(ϕ}i − ϕj)], i, j = 1, 2, 3, i̸= j.

am#biBimiBM; i?2b2 pHm2b BMiQ i?2 2[miBQMb Q7 KQiBQM r2 `2 H2/ iQ i?2 bvbi2K

¨ ϕi = κ3/2 3 ( j=1,j̸=i mjsin(ϕj − ϕi) | sin(ϕj− ϕi)|3 , i = 1, 2, 3.

6Q` `2HiBp2 2[mBHB#`B- i?2 M;mH` p2HQ+Biv Bb i?2 bK2 +QMbiMi 7Q` HH Kbb2b QM i?2 2[miQ`X .2MQiBM; i?Bb p2HQ+Biv #v α ̸= 0- i?2 +?M;2 Q7 i?2 M;H2b +M #2 `2T`2b2Mi2/ b

ϕ1 = αt + a1, ϕ2 = αt + a2, ϕ3 = αt + a3,

r?2`2 t Bb i?2 iBK2 M/ a1, a2, a3 `2 `2H +QMbiMibX h?mb

¨ ϕi = 0. lbBM; i?2 MQiiBQMb s1 := κ3/2_sin(ϕ 1− ϕ2) | sin(ϕ1 − ϕ2)|3 , s2 := κ3/2_sin(ϕ 2− ϕ3) | sin(ϕ2− ϕ3)|3 , s3 := κ3/2_sin(ϕ 3− ϕ1) | sin(ϕ3− ϕ1)|3 , r2 `2 H2/ iQ i?2 H;2#`B+ bvbi2K ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ m1s1− m3s2 = 0 −m2s1+ m3s3 = 0 m2s2− m1s3 = 0, r?B+? ?b BM}MBi2Hv KMv bQHmiBQMb-s1 = m3 m2 γ, s2 = m1 m2 γ, s3 = γ,

kR

r?2`2 γ ̸= 0 Bb  `2H T`K2i2`X >2M+2 r2 ?p2 T`Qp2/ i?2 7QHHQrBM; `2bmHiX h?2Q`2K kX 6Q` 2p2`v +mi2 b+H2M2 i`BM;H2 BMb+`B#2/ BM i?2 2[miQ` Q7 S2

κ- r2 +M

}M/  +Hbb Q7 Kbb2b- m1, m2, m3 > 0- r?B+? B7 TH+2/ i i?2 p2`iB+2b Q7 i?2 i`BM;H2

7Q`K  `2HiBp2 2[mBHB#`BmK i?i `Qii2b QM i?2 2[miQ` rBi? Mv +?Qb2M MQMx2`Q M;mH` p2HQ+BivX

h?2 #B7m`+iBQM Q7 i?2b2 bQHmiBQMb r?2M κ p`B2b rb bim/B2/ BM (N) M/ r2 rBHH MQi T`2b2Mi Bi 7m`i?2`X "mi r2 rBHH mb2 i?2 B/2b BMi`Q/m+2/ #Qp2 iQ +QMbB/2` i?2 +m`p2/ N@#Q/v T`Q#H2Kb BM p`BQmb +QMi2tib 7Q` N = 4, 5, 6X

jXkXk _2HiBp2 2[mBHB#`B QM i?2 2[miQ` 7Q` N = 4

A7 r2 `2bi`B+i i?2 KQiBQM Q7 i?2 7Qm` #Q/B2b iQ i?2 2[miQ` Q7 S2 κ- i?2M

ωi =−κ−1/2, ω˙i = 0, Ωi = κ−1, i = 1, 2, 3, 4,

M/ i?2 2[miBQMb Q7 KQiBQM UjX8V iF2 i?2 bBKTH2 7Q`K

¨ ϕi = κ3/2 4 ( j=1,j̸=i mjsin(ϕj− ϕi) | sin(ϕj − ϕi)|3 , i = 1, 2, 3, 4. UjXeV

6Q` i?2 `2HiBp2 2[mBHB#`B- i?2 M;mH` p2HQ+Biv Bb i?2 bK2 +QMbiMi 7Q` HH Kbb2b- bQ r2 /2MQi2 i?Bb p2HQ+Biv #v α ̸= 0 M/ iF2

ϕ1 = αt + a1, ϕ2 = αt + a2, ϕ3 = αt + a3, ϕ4 = αt + a4, r?2`2 a1, a2, a3, a4 `2 `2H +QMbiMib- bQ ¨ ϕi = 0, i = 1, 2, 3, 4. lbBM; i?2 MQiiBQM s1 := κ3/2<sub>sin(ϕ</sub> 1− ϕ2) | sin(ϕ1 − ϕ2)|3 , s2 := κ3/2<sub>sin(ϕ</sub> 2− ϕ3) | sin(ϕ2− ϕ3)|3 , s3 := κ3/2<sub>sin(ϕ</sub> 3− ϕ1) | sin(ϕ3− ϕ1)|3 , s4 := κ3/2<sub>sin(ϕ</sub> 4− ϕ1) | sin(ϕ4 − ϕ1)|3 , s5 := κ3/2<sub>sin(ϕ</sub> 2− ϕ4) | sin(ϕ2− ϕ4)|3 , s6 := κ3/2<sub>sin(ϕ</sub> 3− ϕ4) | sin(ϕ3− ϕ4)|3 , i?2M i?2 i?2Q`2K +M #2 2tT`2bb2/ b 7QHHQrbc

kk

h?2Q`2K jX  M2+2bb`v +QM/BiBQM i?i i?2 [m/`BHi2`H BMb+`B#2/ BM i?2 2[miQ` Q7 S2 κ

-rBi? i?2 7Qm` Kbb2b m1, m2, m3, m4 > 0 i Bib p2`iB+2b- 7Q`Kb  `2HiBp2 2[mBHB#`BmK Bb

i?i

s1s6+ s3s5 = s2s4.

S`QQ7X q2 Q#iBM 7`QK i?2 2[miBQMb Q7 KQiBQM +Q``2bTQM/BM; iQ ¨ϕi i?i

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ −m2s1+ m3s3+ m4s4 = 0 m1s1− m3s2− m4s5 = 0 −m1s3+ m2s2− m4s6 = 0 −m1s4+ m2s5+ m3s6 = 0. UjXdV

hQ ?p2 Qi?2` bQHmiBQMb Q7 i?2 Kbb2b i?M m1 = m2 = m3 = m4 = 0- i?2 /2i2`KBMMi

Q7 i?2 #Qp2 bvbi2K Kmbi pMBb?- r?B+? Bb 2[mBpH2Mi iQ

s1s6+ s3s5 = s2s4.

h?Bb `2K`F +QKTH2i2b i?2 T`QQ7X

jXkXj 1[mBpH2Mi 2[miBQMb Q7 KQiBQM

AM i?Bb bm#b2+iBQM- r2 Q#iBM MQi?2` 7Q`K Q7 2[miBQMb Q7 KQiBQM BM r?B+? i?2 +iBQM Q7 i?2 BbQK2i`v ;`QmTb i?i /2}M2 i?2 `2HiBp2 2[mBHB#`B Bb +QMb2`p2/X G2i mb MQr BMi`Q/m+2 bQK2 2[mBpH2Mi 2[miBQMb Q7 KQiBQM i?i `2 bmBi#H2 7Q` i?2 FBM/ Q7 bQHmiBQMb r2 `2 b22FBM;X 6B`bi- #v 2HBKBMiBM; ωi 7`QK i?2 +QMbi`BMib ;Bp2M #v 2[miBQM UjX9V r2 ;2i

κ(x2_i + y_i2) + (|κ|1/2_z i+ 1)2 = 1, UjX3V M/ bQHpBM; 2tTHB+BiHv 7Q` zi- r2 Q#iBM zi =|κ|−1/2[ 5 1_{− κ(x}2 i + yi2)− 1]. UjXNV

h?2 B/2 ?2`2 Bb iQ 2HBKBMi2 i?2 7Qm` 2[miBQMb BMpQHpBM; z1, z2, z3, z4- #mi i?2v biBHH

TT2` BM i?2 i2`Kb r2 ij BM i?2 7Q`K σ(zi− zj)2 b σ(zi− zj)2 = κ(x2 i + yi2− x2j − y2j)2 -6 1− κ(x2 i + y2i) + 5 1− κ(x2 j + y2j) .2. UjXRyV

kj

h?2 +b2 Q7 T?vbB+H BMi2`2bi Bb r?2M κ Bb MQi 7` 7`QK x2`Q- bQ i?2 #Qp2 2tT`2bbBQM 2tBbib 2p2M 7Q` bKHH κ > 0 mM/2` i?Bb bbmKTiBQMX h?2M i?2 2[miBQMb Q7 KQiBQM #2+QK2

¨ xi = N ( j=1,j̸=i mj -xj− + 1₋κρ2ij 2 , xi . + 1− κρ2ij 4 ,3/2 ρ3 ij − κ( ˙xi2+ ˙yi2+ κBi)xi ¨ yi = N ( j=1,j̸=i mj -yj − + 1− κρ2ij 2 , yi . + 1₋κρ2ij 4 ,3/2 ρ3 ij − κ( ˙xi2+ ˙yi2+ κBi)yi, UjXRRV r?2`2 ρ2_ij = (xi− xj)2 + (yi− yj)2+ κ(Ai− Aj)2 (√1− κAi+ 6 1− κAj)2 , Ai = x2i + yi2, Bi = (xix˙i+ yiy˙i)2 1_{− κ(x}2 i + y2i) , i = 1, 2, 3, 4.

Ai Bb Q#pBQmb i?i 7Q` κ = 0 r2 `2+Qp2` i?2 +HbbB+H L2riQMBM 2[miBQMb Q7 KQiBQM Q7 i?2 THM` 9@#Q/v T`Q#H2KX HbQ- bBM+2 i?2 `2HiBp2 2[mBHB#`B 7Q` L2riQMBM 2[miBQMb `2 BMp`BMi mM/2` i?2 +iBQM Q7 i?2 `QiiBQM Ki`Bt ;Bp2M #v UjXjV- M/ i?2 Qi?2` i2`Kb BM 2[miBQM UjXRRV /2T2M/ 2bb2MiBHHv QM KmimH /BbiM+2b M/ Bib /2`BpiBp2b- i?2M i?2 +Q``2bTQM/BM; `2HiBp2 2[mBHB#`B 7Q` i?2 M2r bvbi2K Bb +QMb2`p2/ #v UjXjVX

jXkX9 h?2 +b2 Q7 irQ KbbH2bb #Q/B2b 7Q` N = 4

q2 MQr +QMbB/2` i?2 +b2 r?2M irQ Qmi Q7 i?2 7Qm` ;Bp2M Kbb2b `2 KbbH2bb- m3 =

m4 = 0X h?2M i?2 2[miBQMb Q7 KQiBQM #2+QK2

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ¨ x1 = m2 -x2− + 1−κρ212 2 , x1 . + 1−κρ212 2 ,3/2 ρ3 12 − κ( ˙x12+ ˙y12+ κB1)x1 ¨ y1 = m2 -y2− + 1−κρ212 2 , y1 . + 1−κρ212 4 ,3/2 ρ3 12 − κ( ˙x12+ ˙y12+ κB1)y1 UjXRkV

k9 ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ¨ x2 = m1 -x1− + 1−κρ212 2 , x2 . + 1−κρ212 4 ,3/2 ρ3 12 − κ( ˙x22+ ˙y22+ κB2)x2 ¨ y2 = m1 -y1− + 1−κρ212 2 , y2 . + 1−κρ212 4 ,3/2 ρ3 12 − κ( ˙x22+ ˙y22+ κB2)y2 UjXRk#V ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ¨ x3 = m1 -x1− + 1−κρ213 2 , x3 . + 1−κρ213 4 ,3/2 ρ3 13 +m2 -x2− + 1−κρ232 2 , x3 . + 1−κρ223 4 ,3/2 ρ3 23 − κ( ˙x32+ ˙y32+ κB3)x3 ¨ y3 = m1 -y1− + 1−κρ213 2 , y3 . + 1−κρ213 4 ,3/2 ρ3 13 + m2 -y2− + 1−κρ232 2 , y3 . + 1−κρ232 2 ,3/2 ρ3 32 − κ( ˙x32+ ˙y32+ κB3)y3 UjXRk+V ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ¨ x4 = m1 -x1− + 1−κρ214 2 , x4 . + 1−κρ214 4 ,3/2 ρ3 14 + m4 -x4− + 1−κρ242 2 , x2 . + 1−κρ242 4 ,3/2 ρ3 42 − κ( ˙x42+ ˙y42 + κB4)x4 ¨ y4 = m1 -y1− + 1−κρ214 2 , y4 . + 1−κρ214 4 ,3/2 ρ3 14 + m4 -y4− + 1−κρ242 2 , y2 . + 1−κρ242 4 ,3/2 ρ3 42 − κ( ˙x42+ ˙y42+ κB4)y4, UjXRk/V r?2`2 ρ2 ij = ρ2ji, i̸= j-ρ2<sub>ij</sub> = (xi− xj)2+ (yi− yj)2+ κ(x2 i + yi2− x2j − yj2)2 [61<sub>− κ(x</sub>2 i + y2i) + 5 1<sub>− κ(x</sub>2 j + y2j)]2 . q2 +M MQr b?Qr i?i r?2M m1 = m2 =: m > 0 M/ m3 = m4 = 0- b[m`2@HBF2 `2HiBp2

2[mBHB#`B- BX2X 2[mBHi2`H 2[mBM;mH` [m/`BHi2`Hb- Hrvb 2tBbi QM S2 κX

h?2Q`2K 9X AM i?2 +m`p2/ 9@#Q/v T`Q#H2K- bbmK2 i?i m1 = m2 =: m > 0 M/

m3 = m4 = 0X h?2M- BM S2κ- i?2`2 `2 irQ +B`+H2b Q7 `/Bmb 0 < r < κ−1/2- T`HH2H rBi?

i?2 2[miQ`- bm+? i?i  b[m`2 +QM};m`iBQM BMb+`B#2/ BM i?Bb +B`+H2- rBi? m1, m2 i i?2

QTTQbBi2 2M/b Q7 QM2 /B;QMH M/ m3, m4 i i?2 QTTQbBi2 2M/b Q7 i?2 Qi?2`

/B;QMH-7Q`Kb  `2HiBp2 2[mBHB#`BmKX

S`QQ7X P#b2`p2 i?i i?2 p`B#H2 r /2}M2/ #2HQr Bb `2Hi2/ rBi? i?2 ?B;?i 7`QK i?2 2[miQ` iQ i?2 THM2 +QMiBMBM; i?2 +QM};m`iBQM BMb+`B#2/ QM i?2 +B`+H2 Q7 `/Bmb r BM TQbBiBp2 UMQ`i?2`M ?2KBbT?2`2V Q` M2;iBp2 b2Mb2 UbQmi?2`M ?2KBbT?2`2VX aQ- r2 +M bbmK2- rBi?Qmi HQbb Q7 ;2M2`HBiv- i?i i?2 #Q/B2b `2 BM i?2 MQ`i?2`M ?2KBbT?2`2X h?2M

k8

m

2

m

1

m

4

m

3

m

2

m

1

m

3

m

4 6B;m`2 jXR, h?2 +b2 Q7 k 2[mH Kbb2b M/ k KbbH2bb #Q/B2b BM i?2 MQ`i?2`M Q` BM i?2 bQmi?2`M ?2KBbT?2`2

r2 Kmbi +?2+F i?2 2tBbi2M+2 Q7  bQHmiBQM Q7 i?2 7Q`K

q = (q1, q2, q3, q4)∈ S2κ, qi= (xi, yi), i = 1, 2, 3, 4.

x1 = r cos αt, y1 = r sin αt,

x2 =−r cos αt, y2 =−r sin αt,

x3 = r cos(αt + π/2) =−r sin αt, y3 = r sin(αt + π/2) = r cos αt,

x4 =−r cos(αt + π/2) = r sin αt, y4 =−r sin(αt + π/2) = −r cos αt,

r?2`2

x2_i + y_i2 = r2, ρ2 = ρ2₁₃ = ρ2₁₄ = ρ2₂₃ = ρ2₂₄= 2r2, ρ2₁₂= ρ2₃₄= 4r2.

am#biBimiBM; i?2b2 2tT`2bbBQMb BMiQ i?2 bvbi2K UjV- i?2 }`bi 7Qm` 2[miBQMb H2/ mb iQ

α2 ₌ m

4r3_{(1− κr}2₎3/2,

r?2`2b i?2 Hbi 7Qm` 2[miBQMb vB2H/

α2 = 2m(1− κρ2 2 ) ρ3₍₁₋ κρ2 4 )3/2(1− κr2) .

ke

m

1

m

3

m

2

m

4 6B;m`2 jXk, h?2 +b2 Q7 irQ 2[mH Kbb2b M/ irQ KbbH2bb #Q/B2bX

aQ- iQ ?p2  bQHmiBQM- i?2 2[miBQM m

4r3_{(1− κr}2₎3/2 =

2m(1₋ κρ₂2) ρ3₍₁₋ κρ2

4 )3/2(1− κr2)

Kmbi #2 biBb}2/X h?Bb 2[miBQM Bb 2[mBpH2Mi iQ 1 8r3_{(1− κr}2₎3/2 = 1 2√2r3₍₁₋κr2 2 )3/2 , r?B+? H2/b iQ 3κr2 = 2. 6Q` S2 κ- Bi H2/b iQ r =62/3κ−1/2. aBM+2 r < κ−1/2- bm+?  bQHmiBQM Hrvb 2tBbib BM S2

kd

jXkX8 h?2 +b2 Q7 QM2 KbbH2bb #Q/v 7Q` N = 4

G2i m1, m2, m3 = m > 0 M/ bbmK2 i?i m4 = 0X h?2M i?2 2[miBQMb Q7 KQiBQM iF2

i?2 7Q`K ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ¨ x1 = m2 -x2− + 1−κρ212 2 , x1 . + 1−κρ212 4 ,3/2 ρ3 12 + m3 -x3− + 1−κρ231 2 , x1 . + 1−κρ231 4 ,3/2 ρ3 31 − κ( ˙x12+ ˙y12+ κB1)x1 ¨ y1 = m2 -y2− + 1−κρ212 2 , y1 . + 1−κρ212 4 ,3/2 ρ3 12 + m3 -y3− + 1−κρ231 2 , y1 . + 1−κρ231 4 ,3/2 ρ3 31 − κ( ˙x12+ ˙y12+ κB1)y1 UjXRjV ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ¨ x2 = m1 -x1− + 1−κρ212 2 , x2 . + 1−κρ212 4 ,3/2 ρ3 12 + m3 -x3− + 1−κρ232 2 , x2 . + 1−κρ232 4 ,3/2 ρ3 32 − κ( ˙x22+ ˙y22 + κB2)x2 ¨ y2 = m1 -y1− + 1−κρ212 2 , y2 . + 1−κρ212 4 ,3/2 ρ3 12 + m3 -y3− + 1−κρ232 2 , y2 . + 1−κρ232 4 ,3/2 ρ3 32 − κ( ˙x22+ ˙y22+ κB2)y2 UjXRj#V ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ¨ x3 = m1 -x1− + 1−κρ213 2 , x3 . + 1−κρ213 4 ,3/2 ρ3 13 +m2 -x2− + 1−κρ232 2 , x3 . + 1−κρ232 4 ,3/2 ρ3 32 − κ( ˙x32+ ˙y32+ κB3)x3 ¨ y3 = m1 -y1− + 1−κρ213 2 , y3 . + 1−κρ213 4 ,3/2 ρ3 13 + m2 -y2− + 1−κρ232 2 , y3 . + 1−κρ232 4 ,3/2 ρ3 32 − κ( ˙x32+ ˙y32+ κB3)y3 UjXRj+V ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ¨ x4 = m1 -x1− + 1−κρ214 2 , x4 . + 1−κρ214 4 ,3/2 ρ3 14 + m2 -x2− + 1−κρ242 2 , x4 . + 1−κρ242 4 ,3/2 ρ3 42 + m3 -x3− + 1−κρ243 2 , x4 . + 1−κρ243 4 ,3/2 ρ3 43 −κ( ˙x42+ ˙y42 + κB4)x4 ¨ y4 = m1 -y1− + 1−κρ214 2 , y4 . + 1−κρ214 4 ,3/2 ρ3 14 + m2 -y2− + 1−κρ242 2 , y4 . + 1−κρ242 4 ,3/2 ρ3 42 + m3 -y3− + 1−κρ243 2 , y4 . + 1−κρ243 4 ,3/2 ρ3 43 −κ( ˙x42+ ˙y42+ κB4)y4. UjXRj/V

q2 rBHH M2ti b?Qr i?i B7 i?2 MQM@M2;HB;B#H2 Kbb2b `2 2[mH- i?2M i?2`2 2tBbi bQK2 FBi2@b?T2/ `2HiBp2 2[mBHB#`BX

h?2Q`2K 8X *QMbB/2` i?2 +m`p2/ 9@#Q/v T`Q#H2K rBi? Kbb2b m1 = m2 = m3 := m > 0

M/ m4 = 0X h?2M- BM S2κ- i?2`2 2tBbib i H2bi QM2 FBi2@b?T2/ `2HiBp2 2[mBHB#`BmK

k3

m

1

m

2

m

3

m

4 6B;m`2 jXj,  FBi2 +QM};m`iBQM Q7 j 2[mH Kbb2b M/ QM2 KbbH2bb #Q/vX

M2;HB;B#H2 Kbb Bb i i?2 BMi2`b2+iBQM Q7 i?2 2ti2MbBQM Q7 QM2 ?2B;?i Q7 i?2 i`BM;H2 rBi? i?2 +B`+H2 QM r?B+? HH i?2 #Q/B2b KQp2X S`QQ7X q2 rBHH +?2+F  bQHmiBQM Q7 i?2 7Q`K x1 = r cos αt, y1 = r sin αt, x2 = r cos + αt + 2π 3 , , y2 = r sin + αt + 2π 3 , x3 = r cos + αt + 4π 3 , , y3 = r sin + αt + 4π 3 , , x4 = r cos + αt<sub>−</sub> π 3 , , y4 = r sin + αt<sub>−</sub> π 3 , , r?2`2 ρ2₁₂ = ρ2₁₃ = ρ2₂₃= 3r2, ρ2₄₃ = ρ2₄₁= r2, ρ2₂₄= 4r2.

am#biBimiBM; i?2b2 2tT`2bbBQMb BMiQ i?2 #Qp2 bvbi2K- r2 `2 H2/ iQ i?2 +QM+HmbBQM i?i i?2 7QHHQrBM; irQ 2[miBQMb Kmbi #2

biBb}2/-α2 = √ m 3r3₍₁₋3κr2 4 )3/2 , α2 = m 4r3_{(1− κr}2₎3/2 + m r3₍₁₋ κr2 4 )3/2 .

kN `2HiBp2 2[mBHB#`B-1 √ 3(1₋3κr2 4 )3/2 = 1 4(1_{− κr}2₎3/2 + 1 (1₋ κr₄2)3/2.

ai`B;?i7Q`r`/ +QKTmiiBQMb b?Qr i?i r Bb  bQHmiBQM Q7 i?Bb 2[miBQM B7 Bi Bb  `QQi Q7 i?2 TQHvMQKBH P (r) = a24r24+ a22r22+ a20r20+ a18r18+ a16r16+ a14r14+ a12r12+ a10r10+ a8r8+ a6r6+ a4r4+ a2r2+ a0, a24= 6697290145 16777216 κ 12_{, a} 22 =− 2884257825 524288 κ 11_{, a} 20 = 18063189465 524288 κ 10_, a18 =− 4241985935 32768 κ 9_{, a} 16= 21267471735 65536 κ 8_, a14 =− 584429805 1024 κ 7_{, a} 12= 737853351 1024 κ 6_{, a} 10 =− 41995431 64 κ 5_, a8 = 109080063 256 κ 4_{, a} 6 =− 1530101 8 κ 3_, a4 = 446217 8 κ 2_{, a} 2 =−9318κ, a0 = 649

i?i #2HQM;b iQ i?2 BMi2`pH r ∈ (0, κ−1/2<sub>)</sub> 7Q` S2

κX hQ }M/ Qmi B7 r2 ?p2 bm+?  `QQi- r2

KF2 i?2 bm#biBimiBQM x = r2- M/ Q#iBM i?2 TQHvMQKBH

Q(x) = a24x12+ a22x11+ a20x10+ a18x9+ a16x8+ a14x7 + a12x6+

a10x5+ a8x4+ a6x3+ a4x2+ a2x + a0.

"v .2b+`i2bǶb `mH2 Q7 bB;Mb i?2 MmK#2` Q7 TQbBiBp2 `QQib /2T2M/b QM i?2 MmK#2` Q7 +?M;2b Q7 bB;M Q7 i?2 +Q2{+B2Mib- r?B+? BM im`M /2T2M/b QM i?2 bB;M Q7 κX aQ H2i mb /Bb+mbb i?2 irQ +b2b b2T`i2HvX

AM S2

κ- BX2X 7Q` κ > 0- i?2`2 `2 ir2Hp2 +?M;2b Q7 bB;M- bQ Q +M ?p2 ir2Hp2-

i2M-2B;?i- bBt- 7Qm`- irQ- Q` x2`Q TQbBiBp2 `QQib- bQ i?Bb /Q2b MQi ;m`Mi22 i?2 2tBbi2M+2 Q7  TQbBiBp2 `QQiX >Qr2p2`- r2 +M MQiB+2 i?i Q(κ−1

2 ) =−2.4959 < 0 M/ Q(0) = 649 >

0-bQ  `QQi Kmbi 2tBbi 7Q` x ∈ (0, κ−1/2₎- BX2X 7Q` r ∈ (0, κ−1)-  `2K`F i?i T`Qp2b i?2

jy

jXj h?2 +b2 Q7 M2;iBp2 +m`pim`2 7Q` N = 4 QM H

2 κ b r2 ?p2 K2MiBQM2/ BM a2+iBQM jXR- i?2 `2HiBp2 2[mBHB#`B QM H2

κ +M #2 Q7 i?`22

/Bz2`2Mi FBM/b- /2T2M/BM; QM i?2 bT2+BH ;`QmT Q7 BbQK2i`v r?B+? Bb +iBM; QM i?i bm`7+2X AM i?Bb rv r2 +M ?p2 2HHBTiB+- T`#QHB+ Q` ?vT2`#QHB+ `2HiBp2 `2HiBp2X AM (RR) i?2 mi?Q`b T`Qp2 i?2 MQM@2tBbi2M+2 Q7 T`#QHB+ `2HiBp2 2[mBHB#`BX JQ`2 `2+2MiHv-BM (jR)- i?2 mi?Q`b T`Qp2 i?2 MQM@2tBbi2M+2 Q7 TQHv;QMH ?vT2`#QHB+ `2HiBp2 2[mBHB#`BX AM i?Bb b2+iBQM- r2 ?p2 `2bi`B+i2/ Qm` MHvbBb iQ i?2 +b2 Q7 2HHBTiB+ `2HiBp2 2[mBHB#`B QM H2

κ- HbQ FMQrM b ?vT2`#QHB+ 2HHBTiB+ `2HiBp2 2[mBHB#`BX

jXjXR h?2 +b2 Q7 irQ KbbH2bb #Q/B2b QM H

2 κ

Ai Bb FMQrM Ub22 7Q` BMbiM+2 (e)V- i?i 7Q` Mv N ∈ N- m > 0 M/ z > 1- i?2`2 `2 irQ pHm2b Q7 ω- QM2 TQbBiBp2 M/ QM2 M2;iBp2 bm+? i?i- i?2 BbQK2i`v Ki`Bt A(ωt) /2}M2/ #v 2[miBQM UjXjV- ;2M2`i2b `2HiBp2 2[mBHB#`B r?2`2 i?2 Kbb2b `2 HQ+i2/ i i?2 p2`iB+2b Q7  `2;mH` N@;QMX 6Q` i?2b2 `2bQMb M/ i?2 `2bmHib T`Qp2/ BM i?2 T`2pBQmb b2+iBQMX lM2tT2+i2/Hv- i?Bb Bb i?2 +b2X

AM Q`/2` iQ 7+BHBii2 i?2 MQiiBQMb- 7`QK ?2`2 QM r2 rBHH bbmK2- rBi?Qmi HQbb Q7 ;2M2`HBiv- i?i i?2 M2;iBp2 +m`pim`2 Bb 2[mH iQ −1X AM i?Bb bm#b2+iBQM r2 +QMbB/2` i?2 +b2 r?2M m1 = m2 = m > 0 M/ m3 = m4 = 0X q2 Kmbi +?2+F i?2 2tBbi2M+2 Q`

MQM@2tBbi2M+2 Q7  bQHmiBQM Q7 i?2 7Q`K

q = (q1, q2, q3, q4)∈ H2κ, qi= (xi, yi), i = 1, 2, 3, 4.

x1 = r cos αt, y1 = r sin αt,

x2 =−r cos αt, y2 =−r sin αt,

x3 = r cos(αt + π/2) =−r sin αt, y3 = r sin(αt + π/2) = r cos αt,

x4 =−r cos(αt + π/2) = r sin αt, y4 =−r sin(αt + π/2) = −r cos αt,

r?2`2

x2_i + y_i2 = r2, ρ2 = ρ2₁₃ = ρ2₁₄ = ρ2₂₃ = ρ2₂₄= 2r2, ρ2₁₂= ρ2₃₄= 4r2.

jR

2[miBQMb H2/ mb iQ

α2 = m

4r3_{(1 + r}2₎3/2,

r?2`2b i?2 Hbi 7Qm` 2[miBQMb vB2H/

α2 = √ m 2r3_{(1 +} r2

2)3/2

. aQ- iQ ?p2  bQHmiBQM- i?2 2[miBQM

m 4r3_{(1 + r}2₎3/2 = m √ 2r3_{(1 +} r2 2)3/2 , Kmbi #2 biBb}2/X h?Bb 2[miBQM Bb 2[mBpH2Mi iQ

4(1 + r2)3/2 =√2(1 +r 2 2) 3/2_, r?B+? H2/b iQ 3r2 =_−2,

r?B+? Bb  +QMi`/B+iBQMX >2M+2- i?2b2 Q`#Bib /Q MQi 2tBbi QM H2 −1X

jXjXk h?2 +b2 Q7 QM2 KbbH2bb #Q/v QM H

2 −1

G2i m1, m2, m3 = m > 0 M/ bbmK2 i?i m4 = 0- rBi?Qmi HQbb Q7 ;2M2`HBiv- r2 +M

`2bi`B+i Qm` bim/v iQ i?2 mMBi ?vT2`#QHB+ bT?2`2 7Q` M2;iBp2 +m`pim`2X h?2M r2 rBHH +?2+F  bQHmiBQM Q7 i?2 7Q`K x1 = r cos αt, y1 = r sin αt, x2 = r cos + αt + 2π 3 , , y2 = r sin + αt + 2π 3 , x3 = r cos + αt + 4π 3 , , y3 = r sin + αt + 4π 3 , , x4 = r cos + αt<sub>−</sub> π 3 , , y4 = r sin + αt<sub>−</sub> π 3 , , r?2`2 ρ2₁₂ = ρ2₁₃ = ρ2₂₃= 3r2, ρ2₄₃ = ρ2₄₁= r2, ρ2₂₄= 4r2.

jk

am#biBimiBM; i?2b2 2tT`2bbBQMb BMiQ i?2 bvbi2K URkV 7Q` κ = −1 < 0- r2 `2 H2/ iQ i?2 +QM+HmbBQM i?i i?2 7QHHQrBM; irQ 2[miBQMb Kmbi #2

biBb}2/-α2 = √ m 3r3_{(1 +} 3r2 4 )3/2 , α2 = m 4r3_{(1 + r}2₎3/2 + m r3_{(1 +} r2 4 )3/2 .

*QKT`BM; i?2b2 2[miBQMb r2 Q#iBM i?2 +QM/BiBQM 7Q` i?2 2tBbi2M+2 Q7 i?2 FBi2@b?T2/ `2HiBp2 2[mBHB#`B-1 √ 3(1 + 3r₄2)3/2 = 1 4(1 + r2₎3/2 + 1 (1 + r₄2)3/2.

ai`B;?i7Q`r`/ +QKTmiiBQMb b?Qr i?i r Bb  bQHmiBQM Q7 i?Bb 2[miBQM B7 Bi Bb  `QQi Q7 i?2 TQHvMQKBH P (r) = a24r24+ a22r22+ a20r20+ a18r18+ a16r16+ a14r14+ a12r12+ a10r10+ a8r8+ a6r6+ a4r4+ a2r2+ a0, a24= 6697290145 16777216 , a22 = 2884257825 524288 , a20 = 18063189465 524288 , a18= 4241985935 32768 , a16= 21267471735 65536 , a14= 584429805 1024 , a12 = 737853351 1024 , a10= 41995431 64 , a8 = 109080063 256 , a6 = 1530101 8 , a4 = 446217 8 , a2 = 9318, a0 = 649.

aBM+2 HH +Q2{+B2Mib Q7 i?2 TQHvMQKBH P (r) `2 TQbBiBp2b- #v .2b+`i2b `mH2 Q7 bB;Mb- i?Bb TQHvMQKBH /Q2b MQi ?p2 TQbBiBp2 `QQibX h?2`27Q`2 i?2`2 `2 MQ FBi2 bQHmiBQMb 7Q` i?Bb ivT2 Q7 +QM};m`iBQM BM H2

jj

*?Ti2` 9

_2HiBp2 1[mBHB#`B BM h?2 *m`p2/ 8@"Q/v

S`Q#H2K

9XR _2HiBp2 2[mBHB#`B QM i?2 2[miQ` 7Q` 8@#Q/v T`Q#H2K

_2im`MBM; iQ i?2 2[miBQMb Q7 KQiBQM UjXeV QM i?2 2[miQ` Q7 S2

κ- r2 +QMbB/2` i?2 8@#Q/v

T`Q#H2K M/ TH+2 i?2 #Q/B2b i i?2 p2`iB+2b Q7  T2Mi;QM BMb+`B#2/ BM i?2 2[miQ` Q7 S2

κ i?i /Q2bMǶi HB2 rBi?BM Mv b2KB+B`+H2X UA7 i?2 #Q/B2b `2 BM bQK2 ?H7 Q7  ?2KBbT?2`2

i?2v +MMQi 7Q`K `2HiBp2 2[mBHB#`B- b b?QrM BM (3)VX h?2M i?2 2tT`2bbBQMb Q7 si, i =

1, 2, . . . , 10, `2 s1 := κ3/2_sin(ϕ 1− ϕ2) | sin(ϕ1− ϕ2)|3 , s2 := κ3/2_sin(ϕ 1− ϕ3) | sin(ϕ1− ϕ3)|3 , s3 := κ3/2_sin(ϕ 1− ϕ4) | sin(ϕ1− ϕ4)|3 , s4 := κ3/2_sin(ϕ 1− ϕ5) | sin(ϕ1− ϕ5)|3 , s5 = κ3/2_sin(ϕ 2− ϕ3) | sin(ϕ2− ϕ3)|3 , s6 := κ3/2_sin(ϕ 2− ϕ4) | sin(ϕ2− ϕ4)|3 , s7 := κ3/2_sin(ϕ 2− ϕ5) | sin(ϕ2− ϕ5)|3 , s8 := κ3/2_sin(ϕ 3− ϕ4) | sin(ϕ3− ϕ4|3 s9 := κ3/2_sin(ϕ 3 − ϕ5) | sin(ϕ3− ϕ5)|3 , s10:= κ3/2_sin(ϕ 4 − ϕ5) | sin(ϕ4− ϕ5)|3 .

j9

6`QK i?2 2[miBQMb Q7 KQiBQM UjXeV- r2 i?2M Q#iBM i?2 bvbi2K ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ −m2s1− m3s2 − m4s3− s4m5 = 0 m1s1− m3s5− m4s6− m5s7 = 0 m1s2+ m2s5− m4s8− m5s9 = 0 m1s3+ m2s6+ m3s8− m5s10 = 0 m1s4+ m2s7+ m3s9− m4s10 = 0, r?B+? ?b BM}MBi2Hv KMv bQHmiBQMb-m1 = s10s5− s6s9+ s7s8 s1s8− s2s6+ s3s5 γ m2 = s10s2− s3s9+ s4s8 s1s8− s2s6+ s3s5 γ m3 = s10s1− s3s7+ s4s6 s1s8− s2s6+ s3s5 γ m4 = s9s1− s2s7+ s4s5 s1s8− s2s6+ s3s5 γ m5 = γ,

rBi? γ ̸= 0X aQ iQ 7Q`K  `2HiBp2 2[mBHB#`BmK- }p2 #Q/B2b HvBM; i i?2 p2`iB+2b Q7  T2Mi;QM BMb+`B#2/ BM i?2 2[miQ` i?i /Q2bMǶi biv BM Mv b2KB+B`+H2- Kmbi ?p2 Kbb2b b ;Bp2M #Qp2X

9XRXR 1tKTH2 Q7 `2HiBp2 2[mBHB#`B QM i?2 2[miQ` 7Q` 2[mH Kbb2b

q2 rBHH M2ti +QMbi`m+i M 2tKTH2 BM i?2 5@#Q/v T`Q#H2K BM r?B+? 8 #Q/B2b Q7 2[mH Kbb2b KQp2 QM i?2 2[miQ`X

1tKTH2 RX AM i?2 +m`p2/ 8@#Q/v T`Q#H2K BM S2

κ- B7 i?2 T2Mi;QM i r?Qb2 p2`iB+2b i?2

Kbb2b HB2 Bb `2;mH`- i?2M m1 = m2 = m3 = m4 = m5 > 0X

S`QQ7X 6Q` mi > 0, i = 1, 2, 3, 4, 5- H2i i?2 i?2 M;H2b #2ir22M i?2 Kbb2b M/ i?2 x@tBb

pB2r2/ 7`QK i?2 +2Mi2` Q7 i?2 1m+HB/2M THM2 #2,

ϕ1 = 0, ϕ2 = 2π 5 , ϕ3 = 4π 5 , ϕ4 = 6π 5 , ϕ5 = 8π 5 .

j8

m

1

m

2

m

3

m

4

m

5 6B;m`2 9XR, h?2 +QM};m`iBQM Q7  `2;mH` TQHv;QM Q7 8 2[mH Kbb2bX h?2M s1 := −1 sin2(2π₅ ), s2 := −1 sin2(4π₅ ), s3 := −1 sin2(6π₅ ), s4 := −1 sin2(8π₅ ), s5 := −1 sin2(2π₅ ), s6 := −1 sin2(4π₅ ), s7 := −1 sin2(6π₅ ), s8 := −1 sin2(2π₅ ), s9 := −1 sin2(4π₅ ), s10:= −1 sin2(2π₅ ). lbBM; UjXeV- r2 Q#iBM mi = [γ, γ, γ, γ, γ]T, i = 1, 2, 3, 4, 5.

Mv pHm2 Q7 i?2 M;mH` p2HQ+Biv KF2b i?Bb +QM};m`iBQM  `2HiBp2 2[mBHB#`BmKX _2K`FX Ai Bb MQi /B{+mHi iQ b?Qr i?i i?Bb `2bmHi +M #2 2ti2M/2/ iQ Mv Q// MmK#2` Q7 #Q/B2bX A7 i?2 MmK#2` Q7 #Q/B2b Bb 2p2M- bBM;mH`BiB2b Q++m`- bQ bm+? `2;mH` TQHv;QMb +MMQi 7Q`K `2HiBp2 2[mBHB#`BX

9Xk _2HiBp2 1[mBHB#`B BM i?2 +m`p2/ 8@#Q/v T`Q#H2K BM S

2

M/

H

2

q2 +QMbB/2`  8@#Q/v T`Q#H2K QM k@/BK2MbBQMH bm`7+2b Q7 +QMbiMi +m`pim`2 κ- rBi? 7Qm` Q7 i?2 Kbb2b ``M;2/ i i?2 p2`iB+2b Q7  b[m`2 M/ i?2 }7i? Kbb i i?2 MQ`i? TQH2 Q7 i?2 bT?2`2X h?2 }p2@#Q/v b2i mT Bb /Bb+mbb2/ 7Q` κ > 0 M/ 7Q` κ < 0X q?2M i?2 +m`pim`2 Bb TQbBiBp2- Bi Bb b?QrM i?i `2HiBp2 2[mBHB#`B 2tBbib r?2M i?2 7Qm` Kbb2b i i?2 p2`iB+2b Q7 i?2 b[m`2 `2 2Bi?2` 2[mH Q` irQ Q7 i?2K `2 BM}MBi2bBKH bm+? i?i

je

Bi /Q2bMǶi z2+i i?2 KQiBQM Q7 i?2 `2KBMBM; i?`22 Kbb2bX >Qr2p2` rBi? irQ TB`b Q7 Kbb2b i i?2 p2`iB+2b Q7 i?2 b[m`2- MQ `2HiBp2 2[mBHB#`B 2tBbibX AM i?2 ?vT2`#QHB+ +b2-κ < 0- i?2`2 2tBbi irQ pHm2b 7Q` i?2 M;mH` p2HQ+Biv r?B+? T`Q/m+2 M2;iBp2 2HHBTiB+ `2HiBp2 2[mBHB#`B r?2M i?2 Kbb2b i i?2 p2`iB+2b Q7 i?2 b[m`2 `2 2[mHX q2 HbQ b?Qr i?i i?2 bQHmiBQMb rBi? MQM@2[mH Kbb2b /Q MQi 2tBbi BM H2X

G2i qi = (xi, yi, zi) #2 i?2 +QQ`/BMi2b Q7 i?2 TQBMi Kbb mi- biBb7vBM; i?2 +QMbi`BMi

x2 i + yi2+ σz2i = κ−1, r?2`2 σ Bb i?2 bB;MmK 7mM+iBQM σ := ⎧ ⎨ ⎩ +1 for κ > 0, −1 for κ < 0.

h?2 BMM2` T`Q/m+i #2ir22M i?2 p2+iQ`b qi = (xi, yi, zi) M/ qj = (xj, yj, zj) Bb ;Bp2M #v

qij = qi· qj := xixj+ yiyj + σzizj. U9XRV

h?2 /BbiM+2 #2ir22M i?2 #Q/B2b mi M/ mj BM S2κ M/ H2κ Bb /2}M2/ b

qij := [(xi− xj)2+ (yi − yj)2 + σ(zi− zj)2]1/2 U9XkV

7i2`  +?M;2 Q7 +QQ`/BMi2b M/  `2@T`K2i`BxiBQM Q7 iBK2 BMi`Q/m+2/ #v _X J`iőM2x M/ *X aBKƦ (k3)- i?i r2 /Q MQi `2T2i ?2`2- r2 +M bbmK2 rBi?Qmi HQbb Q7 ;2M2`HBiv i?i κ = ±1X

*QMbB/2` }p2 TQBMi Kbb2b- mi > 0, i = 1, 2, 3, 4- r?Qb2 TQbBiBQM p2+iQ`b-

p2HQ+BiB2b-M/ ++2H2`iBQMb `2 ;Bp2M #v

qi = (xi, yi, zi), ˙qi = ( ˙xi, ˙yi, ˙zi), ¨qi = (¨xi, ¨yi, ¨zi), i = 1, 2, 3, 4,

i?2 Kbb m5 Bb }t2/- HQ+i2/ i (0, 0, 1) 7Q` #Qi? bB;Mb Q7 i?2 +m`pim`2X h?2M- b b?QrM

BM (RR)- i?2 2[miBQMb Q7 KQiBQM iF2 i?2 7Q`K

¨ qi = N ( j=1,j̸=i mj -qj− σ(qi· qj)qi . [σ− σ(qi· qj)2]3/2 − σ( ˙qi· ˙qi )qi, i = 1, 2, 3, 4, 5. U9XjV

jd

PM +QKTQM2Mib- i?2 2[miBQMb Q7 KQiBQM M/ i?2 +QMbi`BMib +M #2 r`Bii2M b ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ¨ xi ='N_j=1,j̸=i mj -xj−σ(qi·qj)xi * ) σ−σ(qi·qj)2 *3/2 − σ( ˙qi· ˙qi)xi, ¨ yi ='N_j=1,j̸=i mj -yj−σ(qi·qj)yi * ) σ−σ(qi·qj)2 *3/2 − σ( ˙qi· ˙qi)yi, ¨ zi ='N_j=1,j̸=i mj -zj−σ(qi·qj)zi * ) σ−σ(qi·qj)2 *3/2 − σ( ˙qi · ˙qi)zi, x2 i + y2i + σz2i = σ, xix˙i+ yiy˙i+ σziz˙i = 0, i = 1, ...n. U9X9V

9XkXR h?2 TQbBiBp2Hv +m`p2/ 8@#Q/v T`Q#H2K rBi? QM2 Kbb i i?2 MQ`i?

TQH2 M/ 7Qm` Kbb2b 7Q`K  b[m`2 BM S

2

AM i?2 TQbBiBp2Hv +m`p2/ 8@#Q/v T`Q#H2K- r2 bbmK2 i?i QM2 #Q/v Q7 Kbb m5 Bb }t2/ i

i?2 MQ`i? TQH2 M/ i?2 Qi?2` 7Qm` #Q/B2b Q7 Kbb2b m1, m2, m3,M/ m4 `2 HQ+i2/ i i?2

p2`iB+2b Q7  `QiiBM; b[m`2 r?B+? Bb Q`i?Q;QMH iQ z@tBb M/ T`HH2H rBi? i?2 2[miQ` z = 0- r2 Kmbi 2t+Hm/2 i?2 2[miQ`- bBM+2 ?2`2 r2 ?p2 i?i i?2 Kbb2b `2 HQ+i2/ BM MiBTQ/H TQbBiBQMb- r?B+? +Q``2bTQM/ iQ MiBTQ/H bBM;mH`BiB2b M/ `2 2t+Hm/2/ 7`QK Qm` +QM};m`iBQM bT+2 (Rk)X h?2 #Q/B2b `2 KQpBM; QM i?2 mMBi bT?2`2 S2- r?B+? ?b

+QMbiMi +m`pim`2 R Ub22 6B;m`2 9XkVX

R

m

2

m

1

m

4

m

3

m

5 6B;m`2 9Xk, h?2 +b2 Q7 4 2[mH Kbb2b M/ QM2 #Q/v i i?2 MQ`i? TQH2

j3

G2KK eX AM i?2 +m`p2/ 8@#Q/v T`Q#H2K BM S2- B7 7Qm` #Q/B2b Q7 Kbb2b m

1 = m2 =

m3 = m4 = m`2 HQ+i2/ i i?2 p2`iB+2b Q7  `QiiBM; b[m`2 M/ i?2 }7i? #Q/v Q7 Kbb

m5 Bb }t2/ i i?2 MQ`i? TQH2 (0, 0, 1)- i?2M r2 +M +QKTmi2 i?2 M;mH` p2HQ+Biv Q7 i?2

bvbi2K r?B+? /2T2M/b QM i?2 Kbb2b Q7 i?2 }p2 #Q/B2b M/ i?2 /BbiM+2b #2ir22M i?2KX S`QQ7X q?2M κ = 1- i?2 2[miBQMb Q7 KQiBQM `2 ;Bp2M #v

¨ qi = N ( j=1,j̸=i mj -qj − (qi· qj)qi . [1_{− (q}i· qj)2]3/2 − ( ˙q i· ˙qi)qi, i = 1, 2, 3, 4, 5. U9X8V

q2 `2 BMi2`2bi2/ BM bQHmiBQMb Q7 i?2 7Q`K

q = (q1, q2, q3, q4, q5)∈ S2, qi= (xi, yi, zi), i = 1, 2, 3, 4, 5. ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ x1 = r cos αt, y1 = r sin αt, z1 =± √ 1_{− r}2_, x2 =−r cos αt, y2 =−r sin αt, z2 =± √ 1_{− r}2_, x3 =−r sin αt, y3 = r cos αt, z3 =± √ 1− r2_, x4 = r sin αt, y4 =−r cos αt, z4 =± √ 1− r2_, x5 = 0, y5 = 0, z5 = 1, U9XeV

r?2`2 r /2MQi2b i?2 `/Bmb Q7 i?2 +B`+H2 BM r?B+? i?2 b[m`2 +QM};m`iBQM `Qii2b- M/ α /2MQi2b i?2 M;mH` p2HQ+Biv Q7 i?2 `QiiBQMX

.m2 iQ i?2 bvKK2i`v- rBi?Qmi HQbb Q7 ;2M2`HBiv- r2 rBHH r`Bi2 i?2 2[miBQMb Q7 KQiBQM 7Q` xi M/ B;MQ`2 yiX *QMbB/2` ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ q1· q2 = q3· q4 = 1− 2r2 = A, q1· q3 = q1· q4 = q2· q3 = q2 · q4 = 1− r2 = B, qi· q5 =± √ 1_{− r}2 _{= C, i = 1, 2, 3, 4.} U9XdV

am#biBimiBM; i?2b2 +QQ`/BMi2b M/ 2tT`2bbBQMb BMiQ i?2 2[miBQMb Q7 KQiBQM U9X8V +Q``2bTQM/BM; iQ i?2 xi +QQ`/BMi2- r2 ;2i

¨

jN r?2`2 R =− m 4r3√_B − 2mB [r2_{(2− r}2_)]3/2 ± m5C r3 − r 2_α2_.

q2 ;2i i?2 7QHHQrBM; +QM/BiBQM 7Q` i?2 2tBbi2M+2 Q7 i?2 b[m`2 Tv`KB/ `2HiBp2 2[mBHB#`B BM S2 α2 = m 4r3_{(1− r}2₎3/2 + 2m r3_{(2− r}2₎3/2 ± m5 r3_{(1− r}2₎1/2. U9XNV

1[miBQM U9XNV Bb i?2 M;mH` p2HQ+Biv Q7 i?Bb bvbi2K M/ /2T2M/b QM i?2 7Qm` 2[mH Kbb2b m- i?2 Kbb Q7 i?2 #Q/v i i?2 MQ`i? TQH2- M/ i?2 /BbiM+2b #2ir22M i?2KX h?Bb +QKTH2i2b i?2 T`QQ7 Q7 G2KK eX

G2KK dX 6Q` i?2 Tv`KB/H +QM};m`iBQM ;Bp2M BM G2KK e- r?2M m1 = m2 = m > 0

M/ m3 = m4 = M > 0- i?2 b[m`2 Tv`KB/H `2HiBp2 2[mBHB#`B 2tBbib QMHv B7 m = MX

S`QQ7X h?Bb Bb  +QMb2[m2M+2 Q7 i?2 7+i i?i  `2;mH` nĜ;QM- QM  +B`+H2 T`HH2H iQ i?2 2[miQ`- HH Kbb2b Kmbi #2 2[mHX >2`2 r2 T`2b2Mi i?2 7QHHQrBM; T`QQ7 7Q` Qm` +b2X

G2i m1 = m2 = m M/ m4 = m3 = M #2 i?2 irQ TB`b Q7 Kbb2b i p2`iB+2b Q7 

b[m`2- bm+? i?i m1, m2 `2 i i?2 QTTQbBi2 2M/b Q7 QM2 /B;QMH M/ m3, m4 `2 i i?2

QTTQbBi2 2M/b Q7 i?2 Qi?2` /B;QMH- r?2`2 m5 Bb i i?2 MQ`i? TQH2 (0, 0, 1)X lbBM; i?2

+QM};m`iBQMb ;Bp2M #v U9XeV M/ bm#biBimiBM; i?2b2 +QQ`/BMi2b M/ 2tT`2bbBQMb U9XdV BMiQ i?2 2[miBQMb Q7 KQiBQM U9X8V +Q``2bTQM/BM; iQ x1 M/ x2- r2 ;2i

α2 = m 4r3_{(1− r}2₎3/2 + 2M r3_{(2− r}2₎3/2 ± m5 r3_{(1− r}2₎1/2 h?2 2[miBQM +Q``2bTQM/BM; iQ x3 M/ x4 vB2H/ α2 = M 4r3_{(1− r}2₎3/2 + 2m r3_{(2− r}2₎3/2 ± m5 r3_{(1− r}2₎1/2. hQ ?p2  `2HiBp2 2[mBHB#`B- r2 Kmbi ?p2 m 4r3_{(1− r}2₎3/2 + 2M r3_{(2− r}2₎3/2 = M 4r3_{(1− r}2₎3/2 + 2m r3_{(2− r}2₎3/2. h?Bb Bb 2[mBpH2Mi iQ m(δ− 8γ) 4γδ = M (δ− 8γ) 4γδ , r?2`2 γ = (1 − r2₎3/2 M/ δ = (2 − r2₎3/2X h?Bb 2[miBQM +M #2 biBb}2/ QMHv B7 m = MX

*QMb2[m2MiHv- i?2 bQHmiBQM 7Q` i?2 b[m`2 Tv`KB/H T`Q#H2K BM S2 Kmbi ?p2 2[mH