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n2

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(p+ q)n1 − pn2 ' 0

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H(θ, I) = H0(I) + εH1(θ, I)© ¬

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ν ' 0¬

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ν∂S1

∂θ+

1

2ν ′

(∂S1

∂θ

)2

+ ε A(I∗) cos θ = C© ¬

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ν ′ = ∂2H0

∂I∗ 2 .;I$HMXfxIKJ=J.IQP.P4NSR?TURWVKXZYuI~WYUH4Yz

I~zXlPMH.IsYU]feb^LxYU]lNSJ"j^II∗ = I∗0

H4YU]cbWIν = 0

WIsY'IQyb?YS­ipSN§© ¬´[aN;j^I0P.IKJ"IQP.VKJMXfH4Y1VQNSL1N1P.IKRWj^N1

2ν ′0

(∂S1

∂θ

)2

+ ε A(I∗0 ) cos θ = C© B ¬

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S1 = O(√ε)m

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I∗0 + ∂S1

∂θ

I0IKRH4pSN^Y'IQb?YS­ipSN© B ¬YSViYU?YP4IKRWj^N1

2ν ′0 (I − I∗0 )2 + ε A(I∗0 ) cos θ = C

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H1 =∑

k

Ak(I) cos(k | θ)

D

NSRWj^Iθ, I

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J.IQP.P4NSR?TURWVKXZYz?IKRyH4pSNOI~;XlP.H.Ik1

H4YU]cbWI(k1 | ν) ' 0

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R = m′∑

k

Ak cosφk

© @ ¬

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Ak

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Ak = f

(a

a′, ek3 , e′ k4 , (sin I)k5 , (sin I ′)k6

)

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φk

[ANP.PMbWIKL+Y&NSJMLuYφk = k1λ+ k2λ

′ + k3$ + k4$′ + k5Ω + k6Ω

′

= (k1n+ k2n′)t+ k1ε+ k2ε

′ + k3$ + k4$′ + k5Ω + k6Ω

′

© ¶ ¬G\PXfRH.IKXfJ.NP

ki

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k5 + k6 ≡ par

© ¶¶ ¬NbWI6PMXfeR^Xhg?ViYbWI |k3| + |k4| + |k5| + |k6| ≥ |k1 + k2|

m ³ NSJMH4YURyH.N®NP1VQN;Ig?VKXlIKRyH.IQPAk

P.IKJ4pSNjWYO

(M |k1+k2|

) P.IKRWj^NM = max(e, e′, sin I, sin I ′) ≤ 1.

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(sin I2)n, (sin I ′ 2)nNSb

(sin I sin I ′)n¬m

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Ei

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dEi

dt=

∑

j

Dij∂R

∂cj

© ¶A ¬

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P.IKeb^XfRH.I`NSJMLuYzµ

• H.IKJML1NPP.IQVKb^]ZYUJ.IQPiµYSbWIK]lIQP[?YUJ4YNPb?YUXlPk1 = k2 = 0

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′ r´L$b^XfH.N$[aIQbWIKR?YVQNSL[?YUJ4YSjWY1VQNSLnNSb

n′

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δEi =∑

kk1,k2 6=0

Bk

k1n+ k2n′cosφk

︸ ︷︷ ︸termos periodicos

+∑

kk1=k2=0

(Bk cosφk) t

︸ ︷︷ ︸termos seculares

© ¶AD ¬

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k1 = p + qIk2 = −p[?YUJ4Y-VQIKJMH.NP

k1, k2,I\IKRH4pSNuY$ªJ.IQbWKRWVKXZY

ν =k1n+ k2n′ ' 0

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φk

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ν < O(√m′),

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ν = O(√m′)

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q = k1+k2

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Ak

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q = 2¬`IKRHMJ.Iu±=]fXfRWjWYI ^[^XfH.IKJi

YSP-J.IQP.P.NSR?TURWVKXZYSP D µ ¶ I zµ ©q = 3

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b^LuY&[AIKJMXlNjzXlVKXljWYSj^I\R?YSPVQNSRzgWeb^J4YS­Q°;IQPeIQNSL1IKHMJMXlViYSP"j^N-PMXlPMH.IKLuYzm3noPMH4Y&[AIKJMXlNjzXlVKXljWYSj^I\YU0VQNSLybWI`YSP[AIKJMHMb^JM?YS­Q°IQP²L^HMb?YSPIKRyHMJ.INPVQNSJM[aNP²IKRyxNS]fx;Xlj^NP²H.IKR^t?YUL N0L1IQPML1N0[AIKJM«lNj^N^;YUb^L1IKRyH4YURWj^N&[aNSJMH4YURH.N

Y YUL[^]fXfHMbWj^Ij^IQP4YSPk[aIKJMHMb^JM?YS­Q°;IQPim»LuY J.IQP.P.NSR?TURWVKXZY®rÀjzXfH4Y IQPMH4_ixIK]0P.IÀYSPO[AIKJMHMb^JM?YS­Q°IQP6L ^HMb?YSPYUb^L1IKRH4YSjWYSP [aNSJ´IQPMH4Y0[AIKJMXlNjzXlVKXljWYSj^I=LuYURyH.rKLY&VQNSL1IKRWPMb^J4YU^Xf]fXljWYSj^I`VQNSRyHMJ4Y&XfRWbWKRWVKXZYSP j^IQPMHMJMbWVKHMXfxYSPim

FYU]fxIK"YI~z[^]fXlViYS­ipSN\LuYUXlP|J4YUiNYixIK]^[aIK]ZYI~;XlP.H.IKRWVKXZY=j^I[ANSbWViYSP|J.IQP4P.NSR?TURWVKXZYSPwj^IP.IKeb^RWjWY\NSJ.j^IKLRWNPMXlP.H.IKLuY"P.NS]ZYUJFr²j^I²bWIoIQPMH4YSP|J.IQP.P.NSR?TURWVKXZYSPFP4pSN"ªJ4YSViYSPim ´NSL1NJ4YSViYSPFIKRyH.IKRWj^IKL1NP3bWI²PMb?YIQP.H4YU^Xf]fXljWYSj^Irw[AIQybWIKR?Yzm|noPMH4Y XfRWP.H4YU^Xf]fXljWYSj^I3J.IKRyH.I²vSPNSb^HMJ4YSPNSJ.­iYSPcybWIoYUHMb?YULP4NS^J.IoNPMXlPMH.IKLuYr3j^IKx;XljWY"YSN´YUH.N"j^IbWI NP3H.IKJML1NPoJ.IQP.P4NSR?YURyH.IQPoP4pSN`H.IKJML1NPj^IP.IKeb^RWjWY0NSJ.j^IKLR?YSP3I~^VQIKRyHMJMXlVKXljWYSj^IQP´I R?YSP3XfRWVK]fXfR?YS­Q°;IQPim/;IIQPMH4YSPNSJ.IKL+L$b^XfH.Nk[AIQybWIKR?YSPWRWIQVQIQP4YUJMXZYUL1IKRH.INPH.IKJML1NP\J.IQP.P.NSR?YURH.IQP`jWYSP=P.NS]fbW­Q°;IQP'© ¶AD ¬"P.I$H.NSJMR?YULj^NHMXf[AN I?._ R?pSNP4pSNLuYUXlPjzXfxIKJMeIKRyH.IQPsVQNSL1N YSVQNSRyH.IQVQI VQNSL YSPJ.IQP.P.NSR?TURWVKXZYSPj^I[^JMXfL1IKXfJ4YNSJ.j^IKLmoGybWI YSVQNSRyH.IQVQI ?YSPMXlViYUL1IKRH.Ir ybWI´NPFH.IKJML1NPwJ.IQP.P.NSR?YURyH.IQP3j^I´P.IKeb^RWjWY\NSJ.j^IKLYSViYU?YULP.IKRWj^Nj^IQPM[^J.IKQ«fxIKXlPcJ.IKRyH.IYSNPcH.IKJML1NPcR?pSNJ.IQP.P.NSR?YUH.IQP|j^IoL1IKRWNSJ|NSJ.j^IKLm ³ NSJMH4YURyH.N^[?YUJ4Y"b^LuY J.IQP.P.NSR?TURWVKXZY=j^IP.IKeb^RWjWY'NSJ.j^IKLP4IKJ"NSWP.IKJMxS_ixIK]?YSPI~^VQIKRyHMJMXlVKXljWYSj^IQPI`XfRWVK]fXfR?YS­Q°;IQP"j^IKxIKL+P4IKJ´eJ4YURWj^IQP&©ª[aNSJ"I~^IKL[^]lN^Y'I~^VQIKRyHMJMXlVKXljWYSj^I-j^I0±]fXfRWjWY1r zm ¬m

F«lPMXlViYUL1IKRH.IsY$ªJ4YSVKb^J4Y'j^I\b^LuY&J.IQP.P.NSR?TURWVKXZY'j^I`NSJ.j^IKLPMb^[AIKJMXlNSJ [aN;j^I`P.IKJI~;[^]fXlViYSjWY-[AIK]lN&YUH.N'j^IbWI0IKLb^LuY'J.IQP.P.NSR?TURWVKXZY1j^IsNSJ.j^IKL

q ≥ 2,I~;XlP.H.IKL

q]lNSR^eXfHMbWj^IQP\j^I0VQNSRºb^RW­ipSNOIKRyHMJ.IsN1[aIKJMHMb^JM?YSj^N

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»=L t?YULXf]fH.NSR^XZYURWN$j^I±=RWj^NAEIKJi;r b^Lt?YULXf]fH.NSR^XZYURWN-Y\b^L eJ4YUbuj^I]fXfAIKJ.jWYSj^I"ybWI [ANP.PMbWIY\NSJMLuY

H(θ, I) = aI +b

2I2 + ε (2I)

q

2 cos(qθ)© ¶ ¬

P.IKRWj^Na, b

VQNSRWPMH4YURyH.IQPm N6ViYSP.NOJ.IQP.P.NSR?YURH.I7NOXfRH.IKXfJ.NqrYuNSJ.j^IKL£jWYOJ.IQP.P.NSR?TURWVKXZYkbWI&IQP.H4_OP4IKRWj^N

J.IK[^J.IQP.IKRH4YSjWYzIθr6N¥TUR^eb^]lN¥J.IQP4P.NSR?YURyH.IP.IKRWj^N

qθIQb^XfxSYU]lIKRH.IYSN¥TUR^eb^]lNVKJM«fHMXlVQN

φk

j^IgWR^Xlj^NY[?YUJMHMXfJ1jWY¥I~z[^J.IQP4pSN© ¶ ¬m±+jzXhIKJ.IKRWVKXZY¥IKRyHMJ.I

θIφk

rbWI6N¥[^JMXfL1IKXfJ.Nr6b^LuYÀxSYUJMXZ_ixIK]ViYURHGSR^XlViYIKRWb?YURyH.NN®P.IKeb^RWj^N R?pSN^m±¤YS­ipSN

Irb^LuYb^RW­ipSN®jWYSP6I~^VQIKRyHMJMXlVKXljWYSj^IQPI§jWYSPkXfRWVK]fXfR?YS­Q°IQPim±

J.IK[^J.IQP.IKRH4YS­ipSNj^I\YU]feb^LuYSP J.IQP4P.NSR?TURWVKXZYSPI~;H.IKJMR?YSP0©p < 0

¬²[^J.IQVKXlP4Y&]lIKxYUJ´IKLVQNSRyH4Y&b^LH.IKJML1NYsLuYUXlPRWIQPMH.I`t?YULXf]fH.NSR^XZYURWNOybWIsRWNPR?pSN1VQNSRWPMXlj^IKJ4YUJ.IKL1NP-©´IiYUb^er ¶A@C@ yY¬m

G\P"[aNSRH.NPj^I0IQb^Xf]fXf^JMXlN1j^IQPMH.I0t?YULXf]fH.NSR^XZYURWNOIQPMH4YUJ4pSN1jWYSj^NP"[AIK]ZYSP"IQb?YS­Q°IQPdI

dt= 0

dθ

dt= 0

IKRWb?YURyH.N§ybWIuYSP0ªJ.IQbWKRWVKXZYSP&j^IuNP.VKXf]ZYS­ipSNνIuYIQPMH4YU^Xf]fXljWYSj^Iuj^IQPMH.IQP$[aNSRH.NP0g?ViYUL½j^IgWR^XljWYSP$[AIK]lN

tWIQP.P.XZYURWNν2 = hess(H)IKxYU]fb?YSj^NRWNP&[aNSRH.NP'j^IuIQb^Xf]fXf^JMXlN^m ;I

ν2 > 0N[ANSRyH.NÀj^IuIQb^Xf]fXf^JMXlN§rkIQPMH4_QxIK] IKRWb?YURyH.NÀybWIOP.I

ν2 < 0r0XfRWP.H4_QxIK]m

³ JMXfL1IKXfJ4YUL1IKRyH.I0VQNSRWPMXlj^IKJ.IKL1NP NsViYSP.N$R?pSN$[AIKJMHMb^JM?YSj^N6©ε = 0

¬m3±=PP.NS]fbW­Q°;IQP²eIKJ4YU]lIQP´j^IQPMH.I"t?YULXf]hH.NSR^XZYURWN0IKLVQN;NSJ.j^IKR?YSjWYSP3[ANS]ZYUJ.IQP

k =√

2I cos θ, h =√

2I sin θP4pSN`VK«fJ.VKb^]lNPoVQNSRWVQrKRHMJMXlVQNPIKLH.NSJMRWN

jWY'NSJMXfeIKL+j^IsJ4YUXlNI = cte, I > 0

I\ªJ.IQbWKRWVKXZYSP"VQNSRWPMH4YURH.IQPν = a+ bI

;IaIb[ANP.PMbWIKL NsL1IQPML1N$PMXfeR?YU];YSPoHMJ4Y MIKH.qSJMXZYSP²H.KLN0L1IQPML1N$P4IKRyHMXlj^N$j^I"J.NSH4YS­ipSNk© FXfeb^J4Y ¶ Y¬m N

IKRH4YURyH.N^zP.IaIbH.KLPMXfeR?YU]AjzXhIKJ.IKRH.IzIKRyH4pSN-I~;XlPMH.Ib^LVQIKJMH.N$xSYU]lNSJ´j^I

I = −ab

H4YU]abWIY0ªJ.IQbWKRWVKXZYr\Ryb^]ZYW]lNSeNN1TUR^eb^]lN

θH.NSJMR?Y P4I0J.IQP.P.NSR?YURH.Imo±N'[?YSP4YUJ"j^I`xYU]lNSJ.IQP"j^I

I < −ab

Y-xYU]lNSJ.IQP"j^II > −a

b

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H.IKL1NP\b^LuYOXfRyxIKJ.P4pSN6RWNkP.IKRHMXlj^Nj^I$J.NSH4YS­ipSNjWYSP`P.NS]fbW­Q°;IQPim`± J.IQP.P.NSR?TURWVKXZYkI~WYSVKH4Yug?ViYOj^IQP4VKJMXfH4Yu[ANSJb^LVQNSR¾b^RyH.N1gWR^XfH.N1j^I

2q[aNSRH.NPj^I0IQb^Xf]fXf^JMXlN§© FXfeb^J4Y ¶ ¬m

=b?YURWj^NÀ[AIKJMHMb^JM?YUL1NP1N§[^J.NS^]lIKLuYzYH.NS[aNS]lNSeXZY¥jWYSP'P.NS]fbW­Q°;IQP-L$bWjWYzm ;IaIbH.KL¦NL1IQP.L1N

PMXfeR?YU]R?pSNs[ANj^II~;XlPMHMXfJJ.IQP.P.NSR?TURWVKXZY$I=YSP²P4NS]fbW­Q°IQP´VQNSRyHMXfRb?YULP4IKRWj^N$VK«fJ.VKb^]lNPi;LuYSPYUeNSJ4Y&j^IQPM]lN;ViYSj^NPJ.IQPM[AIQVKH.NujWYONSJMXfeIKL © FXfeb^J4Y Y¬m ;I&NkPMXfeR?YU]wj^I

aIbr$jzXhIKJ.IKRyH.I7IKRH4pSNu[ANj^I-I~zXlPMHMXfJJ.IQP.P4NSR?TURWVKXZY

RWNViYSP4NR?pSN[AIKJMHMb^JM?YSj^N^mn3L¤[^JMXfL1IKXfJ.N§]fb^eyYUJi|xSYUL1NP&VQNSRWPMXlj^IKJ4YUJ-bWIH.IKL1NP&b^LuYJ.IQP.P4NSR?TURWVKXZYj^I[^JMXfL1IKXfJ4YÀNSJ.j^IKL©

q = 1¬m zYUaIKL1NP'bWIOR?YJ.IQP.NSR?TURWVKXZYÀI~WYSVKH4Y

a = −bI 3IkPMb^WPMHMXfHMb^XfRWj^N¥XlP.H.NRWNt?YULXf]fH.NSR^XZYURWNÀ© ¶ ¬ NS^H.IKL1NPH = − b

2I2 + ε

√2I cos θ

N¥b?YU] r6?YSP.XlViYUL1IKRyH.IYÀIQb?YS­ipSNj^N¥[arKRWjzb^]lNPMXfL[^]lIQPm ±\P'P.NS]fbW­Q°IQPj^IQP.H.I6t?YULXf]fH.NSR^XZYURWNR?YSP1VQNUNSJ.j^IKR?YSjWYSP

k, hIQP.H4pSN J.IK[^J.IQP.IKRH4YSjWYSPkR?Y FXfeb^J4Y m ³ N;j^IKL1NP6NSWP.IKJMxSYUJOb^LuYJ.IKeXZpSN j^I]fXf^J4YS­ipSN

©NxSYUXlP¿¬0IKLH.NSJMRWNj^N[ANSRyH.Nj^I1IQyb^Xf]fXf^JMXlN§IQPMH4_QxIK]81

©θ = 0

¬cIJ.IKeXl°;IQP$j^I1VKXfJ.VKb^]ZYS­Q°;IQP-I~;H.IKJMR?YSP&IXfRH.IKJMR?YSPu©IKL£H.NSJMRWNj^I

2

¬m&± ]fXfR^t?Y6[ANSRyHMXf]ft?YSjWYrY6P.IK[?YUJ4YUHMJMXf1bWIP.IK[?YUJ4YYSP`HMJ.QP0J.IKeXl°;IQP0I'[?YSP4Y[AIK]lN-[aNSRH.Nj^I\IQb^Xf]fXf^JMXlN'XfRWPMH4_ixIK].

3

©θ = ±π ¬m ³ IKJ.VQIK?YUL1NP=ybWI0YSPVKXfJ.VKb^]ZYS­Q°;IQPXfRyH.IKJMR?YSP[AN;j^IKLP4I

H.NSJMR?YUJ²IKLNP4VKXf]ZYS­Q°IQPoj^I [AIQb^R?Y0YUL[^]fXfHMbWj^I=YSN`J.IQj^NSJ²j^I 2

P4IKL1IK]ft?YURyH.IQP Y\]fXf^J4YS­Q°IQPiyN0ybWI J.IQVQIKAI"NRWNSL1Isj^Is]fXf^J4YS­Q°IQP[?YUJ4YSj^NQ~zXlViYSP'© ?IKJMJ4YUK§IK]f]lNOI ;IQP.PMXfR¬ NSb]fXf^J4YS­Q°IQP=j^I0[aIQbWIKR?YSPI~^VQIKRyHMJMXlVKXljWYSj^IQP© \J.IQIKRaIKJMe¬m²±]lrKLjzXlP.P.N^[aN;j^IKLYSVQNSRyH.IQVQIKJ]fXf^J4YS­Q°;IQP´j^I`YUL[^]fXfHMbWj^I`L$b^XfH.N-eJ4YURWj^I`bWI\YSViYU?YULXfRzVK]fb^XfRWj^NuY'NSJMXfeIKL?P.IKRWj^N'[aNSJMH4YURH.N1P.IKL1IK]ft?YURH.IQP=Y'VKXfJ.VKb^]ZYS­Q°;IQPiWNbWI0r0VQNSR^tWIQVKXlj^NuVQNSL1NVKXfJ.VKb^]ZYS­Q°;IQP[?YUJ4YSj^Ni~;XlViYSPim

NViYSP.Nj^Ib^LuYJ.IQP.P.NSR?TURWVKXZYj^I1P4IKeb^RWjWYNSJ.j^IKL ©q = 2

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θ = π2, 3π

2

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φk . /)' . $0$'1203+465#78:9/;(<=#">+?:@AB"0 '( . 180C!'#">+?:@DE" ' . 8

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q = 1©ª[ArKRWjzb^]lNy¬m

FXfeb^J4Y D µ/;NS]fbW­Q°;IQPj^Nt?YULXf]fH.NSR^XZYURWNOj^Is±=RWj^NAEIKJ"RWN1ViYSP4N'[aIKJMHMb^JM?YSj^NuJ.IQP.P.NSR?YURyH.IW[?YUJ4Yq = 2

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φ = (p+ q)λ1 − pλ2 + (r − q)$2 − r$1

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d2φ

dt2= (p+ q)

d2λ1

dt2− p

d2λ2

dt2+ (r − q)

d2$2

dt2− r

d2$1

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d2φ

dt2= A e2 sinφ

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dt2=B

e2cosφ sinφ

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φ = (p+ q)λ1 − pλ2 + (r − q)Ω2 − rΩ1

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¶AD

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φ = 4λ1 − 2λ2 − Ω1 − Ω2VKbMNOVQN;Ig?VKXlIKRyH.Ir-[^J.NS[ANSJ.VKXlNSR?YU]²Ysin I1 sin I2

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φ1 = 4λ1 − 2λ2 − 2Ω1

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1

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2

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φ?[aNSXlP\NPRWN;j^NP

Ωi

[^J.IQVQIQP4YULdj^IKxXlj^Nu[aNSJ\I~zIKL[^]lN^7YSNkYSV.t?YUH4YUL1IKRyH.Nj^N-[^JMXfLuYUJMXlN ^LuYSPIQPMH4Y&xYUJMXZYS­ipSN-R?pSNr=H4pSN-J4_U[^XljWY'VQNSL1N'[?YUJ4Y-bWI\P4IIMYUL MNSeyYSj^NP´NSJ4Y ¾b^RyH.NVQNSLNPH.IKJML1NP"j^I$VKb^JMH.N[aIKJM«lN;j^N^m3·`IKxIKL1NP"]lIKL&^J4YUJH4YUL$ArKL¼bWIWYUXfRWjWYP.IKRWj^Nuj^I0VKb^JMH.N[aIKJM«lN;j^N^WI~zXlPMH.IKLH.IKJML1NP`j^I&[^JMXfL1IKXfJ4Y6NSJ.j^IKLuYuR?YSP`I~^VQIKRyHMJMXlVKXljWYSj^IQPsbWI&R?pSNkP4IKL[^J.I&P4pSNkj^IQPM[^J.IKQ«fxIKXlPim$G YUH.NOj^I-H.IKJbWI"VQNSRWPMXlj^IKJ4YUJ´xSYUJMXlNP²H.IKJML1NP´R?Y\ªb^RW­ipSN&[AIKJMHMb^JM?YSj^NSJ4Y0YUVQNSLybWI"N$HMJ4YUH4YUL1IKRyH.NYUR?YU]f«fHMXlVQN-VK]Z_SP.PMXlVQNP.II.YOL$b^XfH.NVQNSL[^]fXlViYSj^N§IIKLeIKJ4YU]|YSP0IQb?YS­Q°;IQP0R?pSN[ANj^IKLP.IKJ`J.IQP.NS]fx;XljWYSPsj^IKxIKRWj^NU P.IuJ.IQVKb^JMJMXfJ&YL1IKH.N;j^NS]lNSeXZYSP"Rb^L1rKJMXlViYSPim

noLP.IKeb^RWj^N]fb^eyYUJi`j^IKxIKL1NP]lIKxSYUJIKLVQNSRH4YN YUH.Nj^I¥bWIN®VQNSL[ANSJMH4YUL1IKRyH.N jWY®xSYUJMXZ_ixIK]J.IQP.P4NSR?YURyH.I"j^IK[AIKRWj^I NSJMH.IKL1IKRyH.I\j^N`[^]ZYURWN$j^I J.IIKJ.KRWVKXZYsbWIP.IIMY\b^HMXf]fXfYSj^N$[?YUJ4Y`L1IQjzXfJ´YSPo]lNSR^eXfHMbWj^IQPj^NPRWNj^NP I`YSP´XfRWVK]fXfR?YS­Q°IQPim3noL eIKJ4YU]zP.I\N&[aIKJMHMb^JM?YSj^NSJr=L$b^XfH.N'LuYUXlP´LuYSPMXfxN-ybWI\N$[AIKJMHMb^JM?YSj^N^^N[^J.NS^]lIKLuY0g?ViY0L$b^XfH.N&PMXfL[^]fXhg?ViYSj^N$P4I"IQP.NS]ftWIKL1NP´VQNSL1N0[^]ZYURWN$j^IJ.IIKJ.KRWVKXZY-Y0qSJM^XfH4Ysj^N0[^JMXfL1IKXfJ.N^m NIKRH4YURyH.N^?P4I`YSP"LuYSP4YSP NSJ.IKLVQNSL[?YUJ4_QxIKXlPiN'[^J.NS^]lIKLuY-g?ViY'PMXfL[^]fXhg?ViYSj^N1IQP.VQNS]ftWIKRWj^NuVQNSL1N[^]ZYURWN1j^IJ.IIKJ.KRWVKXZY1YSbWIK]lIsj^IgWR^Xlj^N1[AIK]lNxIQVKH.NSJ"L1NSL1IKRH.NkYUR^eb^]ZYUJ"H.NSH4YU]|j^NPj^NSXlPVQNSJM[ANP$© \J.IQIKRaIKJMe ¶A@ ¬m NP`j^NSXlP0ViYSP.NPi7r-[aNP4PM«fxIK]wb^HMXf]fXfYUJsNOHMJ4YUH4YUL1IKRH.NVK]Z_SP.PMXlVQN6VQNSL¤YSP0IQb?YS­Q°IQP`j^I ¢ YUeJ4YUR^eII'V4tWIKeyYUJYP4NS]fbW­Q°IQP=NSRWj^I-Y']lNSR^eXfHMbWj^I-j^I$VQNSR¾b^RW­ipSNO[aNP.P.bWI0b^L¼VQIKRHMJ.NOj^I0]fXf^J4YS­ipSNkIQPMH4_ixIK]wY @ j^NP"RWN;j^NPL ^HMbWNPim

FXfR?YU]fL1IKRH.Ioj^IKxIKL1NP-]lIKL$^J4YUJybWIuRyb^LuYJ.IQP.P.NSR?TURWVKXZY§j^IuHMXf[ANXfRWVK]fXfR?YS­ipSN¥N§IIKXfH.NÀj^N§YSV4t?YUH4YL1IKRH.NÀj^N§[^JMXfLuYUJMXlN¥ruL&b^XfH.N¥XfL[aNSJMH4YURH.Im ´NSL1N M_L1IKRWVKXlNSR?YUL1NPi²N¥YSV.t?YUH4YUL1IKRH.NruJ.IQP.[aNSRWP4_ixIK][AIK]ZY'[^J.IQVQIQP4pSNOj^NP"RWN;j^NPjWYSPqSJM^XfH4YSP?I0XfRWbWI$RWNuVQNSL[ANSJMH4YUL1IKRyH.N6j^NP=YUJMeb^L1IKRH.NP\j^N1HMXf[aN

φ1, φ2

mGYSV4t?YUH4YUL1IKRyH.NkVQNSRyHMJMXf^bWIsH4YUL&arKLIKL[^JMXfxXf]lIKeXZYUJ\VQIKJMH.NP[^]ZYURWNPj^IsJ.IIKJ.KRWVKXZYzWVQNSL1N1[aNSJ"I~^IKL[^]lNN'[^]ZYURWN1IQb?YUH.NSJMXZYU]cj^N[^JMXfLuYUJMXlN^m

»=LI~zIKL[^]lNj^I`J.IQP4P.NSR?TURWVKXZY-HMXf[aN'XfRWVK]fXfR?YS­ipSNur`Y-J.IQP4P.NSR?TURWVKXZY µ ¶ IKRyHMJ.IsNPP¿YUH.rK]fXfH.IQP"j^I zYUHMb^JMRWN^§XfLuYSPOI cIKHMXlPmn²PMH4Y J.IQP.P.NSR?TURWVKXZY P.I§ViYUJ4YSVKH.IKJMXfY®[AIK]ZY ]fXf^J4YS­ipSN®jWY ]lNSR^eXfHMbWj^I¥j^I¥VQNSRºb^RW­ipSNIKLH.NSJMRWNjWYOL1rQjzXZY6jWYSP`]lNSR^eXfHMbWj^IQPsj^NP0RWN;j^NPiAL1IQjzXljWYSPsIKLdJ.IK]ZYS­ipSNYSN6[^]ZYURWNIQb?YUH.NSJMXZYU]oj^I zYUHMb^JMRWN^m¥YUH.IKLuYUHMXlViYUL1IKRyH.IXlPMH.Nr0IQb^XfxSYU]lIKRH.I&Y'jzXfiIKJbWIsN1TUR^eb^]lN

φ = 2(2λ1 − λ2) − Ω1 − Ω2]fXf^J4Y1YSN1J.IQj^NSJ=j^I zWNSbÀP.II.Y©2λ1 − λ2

¬´]fXf^J4YuYSNJ.IQj^NSJ=j^IΩ = Ω1+Ω2

2

m ³ YUJ4Yub^L+IQPMHMbWj^Nuj^IKH4YU]ft?YSj^Nj^IQPMH4Y-J.IQP4P.NSR?TURWVKXZY'xIKJ"[ANSJ"I~zIKL[^]lN ¢ YUQYUJ.N§© ¶A@CBC@ ¬m !"$#&%('69<' 5$9('"#&%<' /'

·`IgWRWI P.IsVQNSL1N1J.IQP.P.NSR?TURWVKXZY']ZYU[^]ZYSVKXZYUR?YOY-J.IK]ZYS­ipSNn1 − 3n2 + 2n3 ' 0

I~zXlPMH.IKRWIKH.I-IKRyHMJ.INP=L1Nix;XfL1IKRyH.NP0L1rQjzXlNP0j^I&HMJ.QP`VQNSJM[aNP`IKL£qSJM^XfH4YkIKL¼H.NSJMRWN6j^I&b^L[^JMXfLuYUJMXlN^m NPMXlP.H.IKLuY&P4NS]ZYUJ"I~;XlPMH.IKLj^NSXlP"P.XlPMH.IKLuYSPj^I0P4YUH.rK]fXfH.IQPbWIsYU[^J.IQP.IKRyH4YUL+IQP.H4Y$J.IQP4P.NSR?TURWVKXZYzµoNPP4YUH.rK]fXfH.IQP"¹ºN^n3b^J.NS[?Y1I0YUR^XfL1IQj^IQP=j^I ^[^XfH.IKJi?I0NP"P4YUH.rK]fXfH.IQP XfJ4YURWjWYz?±JMXlIK]Is»=L$^JMXlIK]|j^Is»J4YURWN^m

¶

GTUR^eb^]lN1J.IQP.P.NSR?TURyH.I&IKRyH4pSN^r0j^IgWR^Xlj^N1VQNSL1Nφ = λ1 − 3λ2 + 2λ3

IRWN$ViYSP.Nsj^NP²P¿YUH.rK]fXfH.IQP²j^I ^[^XfH.IKJ IQPMH.ITUR^eb^]lN-YU[^J.IQP.IKRyH4Ysb^LuY0]fXf^J4YS­ipSN&VQNSL [AIQybWIKR?YsYUL[^]fXfHMbWj^I\IKLH.NSJMRWN"j^I

180mcnoPMH.I3YUH.N XfL[^]fXlViYbWIoP.IKL[^J.I²ybWI 0YUR^XfL1IQj^IQP|I3n3b^J.NS[?Y"IQP.HMXfxIKJ.IKLIKLVQNSR¾b^RW­ipSN$©IKL

J.IK]ZYS­ipSN'Y^[^XfH.IKJ¿¬zXlP.H.N$r;yb?YURWj^Nλ2 = λ3

¹ N&IQPMH4YUJ4_$IKLNS[ANPMXl­ipSN^y[ANSXlPλ1 −λ2 = λ1 −λ3 ' 180

mFYUL$ArKL?j^IKx;Xlj^NuYSN'YUH.N1j^IsbWI0H.NSLuYSj^NP=YSNP[?YUJ.IQPNPL1NxXfL1IKRH.NP=L1rQjzXlNP=IQPMH4pSN1IKLJ.IQP4P.NSR?TURWVKXZY µ ¶ z[ANj^IKL1NPI~;[^J.IQP4YUJ\Y-xSYUJMXZ_ixIK]cJ.IQP.P.NSR?YURH.IsVQNSL1N

φ = (2λ3 − λ2) − (2λ2 − λ1)

I[aN;j^IKL1NP²xIKJ´ybWI\YSP²]lNSR^eXfHMbWj^IQPj^I=VQNSRºb^RW­ipSN-j^NP[?YUJ.IQP´IQP.H4pSN$P.IK[?YUJ4YSjWYSP j^I ¶AB mwnoP.H.I"YUH.N^y[AIKJ¾LXfH.I$YUiIKJ\b^LuYkYU[^J.NQ~zXfLuYS­ipSNORWNkj^IQP.IKRxNS]fxXfL1IKRH.NjWYb^RW­ipSNO[aIKJMHMb^JM?YSj^NSJ4Yz7VQNSRWPMXlj^IKJ4YURWj^NYU[AIKR?YSPNPH.IKJML1NPybWI0[ANP.PMbWIKL+YUJMeb^L1IKRH.NP=j^N'HMXf[AN

(2λ3 − λ2)(2λ2 − λ1)

PMXfL[^]fXhg?ViYURWj^NU P4IÀYSP.P.XfL P.IKb IQP.HMbWj^N^m ³ YUJ4Y IQPMHMbWjWYUJON VQNSL[aNSJMH4YUL1IKRH.NjWYxYUJMXZ_QxIK]\J.IQP.P.NSR?YURyH.I§RWNH.IKL[AN^^j^IKxIKL1NPj^IKH.IKJMLXfR?YUJ\PMb?Y'xYUJMXZYS­ipSN^WNSbP.II.Y

d2φ

dt2=d2λ1

dt2− 3

d2λ2

dt2+ 2

d2λ3

dt2I[?YUJ4Y=XlPMH.N\b^HMXf]fXfYUL1NPoYSPFIQb?YS­Q°;IQPw[^]ZYURWIKH4_UJMXZYSPoj^I ¢ YUeJ4YUR^eIm ?YUiIKRWj^NsYSP3YU[^J.NQ~zXfLuYS­Q°IQPoIL$bWjWYURW­iYSPj^IxYUJMXZ_QxIKXlP YSj^IQb?YSjWYSP\© \J.IQIKRaIKJMe ¶A@ ¢ YUQYUJ.N ¶A@CBC@ ¬;V.tWIKeyYUL1NP v0IQb?YS­ipSN$j^N$[arKRWjzb^]lN&PMXfL[^]lIQP

d2φ

dt2= C sinφ

I`H.IKL1NP=ybWI0N1TUR^eb^]lNφ]fXf^J4YIKLH.NSJMRWNuj^I ¶AB VQNSL[AIKJM«lNj^N1j^IsYU[^J.Ni~;XfLuYSjWYUL1IKRH.I jzXZYSPim

NsViYSP.N$j^NP²P4YUH.rK]fXfH.IQP´j^I"»=J4YURWN^N$TUR^eb^]lNφR?pSNs]fXf^J4YsLuYSP²VKXfJ.VKb^]ZYzm IQPMH.I"ViYSP.N^RWNsIKRyH4YURH.N^;R?pSN

I~zXlPMH.IKL J.IQP4P.NSR?TURWVKXZYSP²L ^HMb?YSP IKRyHMJ.I`NPo[?YUJ.IQP j^IP¿YUH.rK]fXfH.IQPiYUH.N$IQP.H.IbWI"R?pSN&[aIKJMLXfH.I=IIKHMb?YUJYU[^J.NQ~;XfLuYS­Q°;IQP"RWNP\j^IQP.IKRyxNS]fx;XfL1IKRyH.NP0jWYªb^RW­ipSNu[aIKJMHMb^JM?YSj^NSJ4YO[ANSXlP"R?pSNOI~;XlPMH.IKL¼H.IKJML1NP[^J.IK[ANSRWj^IKJ4YURyH.IQPY-[^JMXlNSJMXm´NSRWP.Xlj^IKJ4YURWj^NOY&b^RW­ipSN[aIKJMHMb^JM?YSj^NSJ4YuIKL+PMb?Y&NSJMLuY'LuYUXlPeIKJ4YU]

R =∑

j

∑

k

Ajk cos(k | λ)

NSRWj^I(k | λ)

J.IK[^J.IQP.IKRH4YN¥[^J.N;jzb^H.NIQP.ViYU]ZYUJ1IKRyHMJ.INPxIQVKH.NSJ.IQP(k1, k2, k3)

I(λ1, λ2, λ3)

IKRH4pSN YxYUJMXZYS­ipSN1j^N1TUR^eb^]lNJ.IQP.P4NSR?YURyH.Is[AN;j^I P.I0IQP.VKJ.IKxIKJ"R?Y&NSJMLuY

d2φ

dt2=

∑

k

Ck sin(kφ)

;IVQNSRWPMXlj^IKJ4YUL1NP'YU[AIKR?YSPsNH.IKJML1Nk = 1

H.IKL1NPicRWNxSYUL1IKRH.IOYIQb?YS­ipSNj^N[ArKRWjzb^]lN^cP4qybWIRWIQP.H.IViYSP.N-N'TUR^eb^]lN

φVKXfJ.VKb^]ZY-VQNSL b^L[AIKJM«lNj^N'j^I`YU[^J.NQ~zXfLuYSjWYUL1IKRyH.I ¶A YURWNP0© \J.IQIKRaIKJMe ¶A@ ¢ YUQYUJ.N

¶A@CBC ¬m %<;= C #2 % C'"%<? - !"/#&%<' %<4$%<#2'"nw~zXlPMH.IKL+YU]feb^RWP"VQNSJM[ANPRWN1P.XlPMH.IKLuY'P.NS]ZYUJ"bWI0IQPMH4pSNVQNSRzgWR?YSj^NPIKLJ.IQP4P.NSR?TURWVKXZY ¶ µ ¶ j^I`L1Nix;XfL1IKRyH.NPL1rQjzXlNPiaNSb§P4IIMYzaNuVQNSJM[AN1[AIKJMHMb^?YSj^N6I$N1[AIKJMHMb^JM?YSj^NSJ0P.IKebWIKLaIKL¼L1rQjzXZYzAY1L1IQP.LuY1qSJM^XfH4YzAj^NSRWj^IN&RWNSL1I`j^I=L1NxXfL1IKRH.N1VQN;NSJM^XfH4YU]mG\P´LuYUXlP HMXf[^XlVQNPI=VQNSR^tWIQVKXlj^NP"j^IQP.H.IQP´VQNSJM[ANPP4pSN-NP"YSPMH.IKJ.qSXlj^IQPjWYYULXf]fXZY"j^NPJ.NSXZYURWNPybWIP.IKebWIKLY"L1IQPMLuY=qSJM^XfH4Yj^I ^[^XfH.IKJim §IKRWNPFVQNSR^tWIQVKXlj^NP3P4pSN=NPcHMJ.NSXZYURWNPwj^I

¶

¥YUJMH.I´J.IQVQIKRyH.IKL1IKRyH.I=j^IQP.VQNSAIKJMH.NP3I YU]feb^RWP3P¿YUH.rK]fXfH.IQPwL1IKRWNSJ.IQP3IKL L1Nix;XfL1IKRyH.NsVQN;NSJM^XfH4YU]^VQNSLP4YUH.rK]fXfH.IQPLuYUXlNSJ.IQP=VQNSL1N1·\XlNSRWI0NSb cIKHMXlPim

GYUJMeb^L1IKRH.NuJ.IQP4P.NSR?TURyH.I0RWIQP.H.IsViYSP.N^z[ANP.PMbWIsY&NSJMLuY'eIKJ4YU]φ = p(λ1 − λ2)

I6YNSJ.j^IKL¦jWYJ.IQP.P.NSR?TURWVKXZYÀr zwLuYSPNP&j^IQP.IKRxNS]fxXfL1IKRH.NPuVK]Z_SP.P.XlVQNP-jWYb^RW­ipSN§[AIKJMHMb^JM?YSj^NSJ4Y§R?pSN[ANj^IKL¨LuYUXlPP.IKJ1YU[^]fXlViYSj^NP1YSNIQPMHMbWj^N¥j^IQPMH.IkHMXf[ANj^IkJ.IQP.P.NSR?TURWVKXZYSPi M_§ybWIY§J4YUpSNj^NPP.IKLXlIKXh~^NPLuYUXlNSJ.IQP j^Ns[AIKJMHMb^JM?YSj^N'IN&[aIKJMHMb^JM?YSj^NSJ´R?pSN-r[AIQybWIKR?YsLuYSP PMXfL [^J.Ni~;XfLuY-jWY0b^R^XljWYSj^Im²noPMH.IQP´ViYSP.NPP4pSN'IQPMHMbWjWYSj^NP"YUR?YU]fXfHMXlViYUL1IKRH.I&YUHMJ4YixrQPjWY&H.IQNSJMXZY'VK]Z_SP.PMXlViY-VQNSR^tWIQVKXljWYVQNSL1N'[^J.NS^]lIKLuY'j^I`HMJ.QPVQNSJM[ANPJ.IQPMHMJMXfH.N© ÀNSb^]fH.NSR ¶A@z¶ J.NSbIKJ > ]lIKL1IKRWVQI ¶A@ ¶ ¬m

IQP.P.I'[^J.NS^]lIKLuYzFVQNSRWPMXlj^IKJ4Y P.IONPMXlPMH.IKLuYVQNSRWP.HMXfHMb^«lj^N[ANSJ$j^NSXlPsVQNSJM[ANPsLuYSPMXfxNP&j^ILuYSP4YSPM

ImWL1I~^rKRWj^NP.I-IKL¼NSJM^XfH4YSP`VKXfJ.VKb^]ZYUJ.IQP`I&VQNS[^]ZYUR?YUJ.IQPm NuViYSP.NuLuYUXlP\VQNSL&b^L¼RWNkPMXlPMH.IKLuYuP.NS]ZYUJ\H.IKL1NP

M m©ª[aNSJ1I~^IKL[^]lN*;NS]I6[^]ZYURWIKH4Y NSb [^]ZYURWIKH4YIP¿YUH.rK]fXfH.I¬m»=L H.IKJ.VQIKJuVQNSJM[ANR?pSNLuYSPMXfxNP4I

L1I~^I'P.NS¥Y6YS­ipSNj^Nk[ANSH.IKRWVKXZYU]3eJ4YQx;XfH4YUH.NSJMXlNVQNSRºb^RH.Nj^NP0NSb^HMJ.NP0j^NSXlPims»=HMXf]fXfYURWj^N6b^LPMXlPMH.IKLuYkj^IJ.IIKJ.KRWVKXZY6IKL£J.NSH4YS­ipSN^H4YU]obWINP0VQNSJM[aNPsLuYSPMXfxNPsN;VKb^[aIKLd[ANP.PMXl­Q°;IQP\g^~^YSPi7I'[ANP.PM«fxIK]oYSV4t?YUJsb^LuYXfRH.IKeJ4YU]j^N0L1NxXfL1IKRH.NO©IKRWIKJMe«ZY0RWN$PMXlPMH.IKLuY`J.NSH4YURH.I¬VQNSR^tWIQVKXljWY$VQNSL1NsVQNSRWPMH4YURyH.I=j^I yYSVQNS^X

CbWI

[AIKJMLXfH.I XfRzIKJMXfJoVQIKJMH4YSPViYUJ4YSVKH.IKJMXlPMHMXlViYSP²j^N`L1NxXfL1IKRH.N$jWY\[?YUJMHM«lVKb^]ZY`R?pSNsLuYSPMXfxSYzmFn3L[?YUJMHMXlVKb^]ZYUJiy[?YUJ4YjzXhIKJ.IKRyH.IQP-xYU]lNSJ.IQP&jWYVQNSRWPMH4YURyH.Ikj^IYSVQNS^X`©bWIuIKx;Xlj^IKRyH.IKL1IKRH.Ij^IK[aIKRWj^IKL¨jWYSP&VQNSRWjzXl­Q°;IQP&XfR^XlVKXZYUXlPj^Nk[^J.NS^]lIKLuY¬cI~zXlPMHMXfJ4pSN6J.IKeXl°;IQP0j^N6IQPM[?YS­QNNSRWj^INkL1Nix;XfL1IKRyH.NjWYO[?YUJMHM«lVKb^]ZYr-[AIKJMLXfHMXlj^NI'NSb^HMJ4YSPNSRWj^I=IQP.P.I"L1NxXfL1IKRH.Nr"[^J.NSXf^Xlj^N^m ³ YUJ4Y-VQIKJMH.NPxYU]lNSJ.IQP´j^I

CIQPMH4YSP²J.IKeXl°;IQP²g?ViYULJ.IQjzb^QXljWYSPY0[ANSRyH.NP

j^I0IQb^Xf]fXf^JMXlN^^ybWI`RWNP.XlPMH.IKLuY-J.NSH4YURyH.I$YSj^NSH4YSj^Ng?ViYULJ.IK[^J.IQP.IKRyH4YSj^NP[aNSJ[ANSRyH.NPg^~zNPi^I\J.IQVQIKAIKL+NRWNSL1I´j^I´[aNSRH.NP|]ZYUeJ4YUR^eIiYURWNPm3±[aNP.Xl­ipSN\YU[^J.Ni~;XfLuYSjWY`j^IQP.P4IQPc[ANSRyH.NP3j^I´IQyb^Xf]fXf^JMXlN`r IQP.bWIKLuYUHMXfYSjWYR?Y FXfeb^J4Y B m

|Xfeb^J4Y B µ ³ NPMXl­Q°;IQPj^NP"[aNSRH.NP"]ZYUeJ4YUR^eIiYURWNP=Ryb^L+PMXlP.H.IKLuY'J.NSH4YURyH.Im

¶

G\P`[ANSRyH.NPs]ZYUeJ4YUR^eIiYURWNPL1, L2

IL3

bWI-g?ViYULP.NS^J.I'NIKXh~zN6bWI-b^RWINP0VQNSJM[aNP`LuYSPMXfxNPP4pSNXfRWPMH4_ixIKXlPiWIKRWb?YURyH.N1bWI0NP[ANSRyH.NP

L4, L5

bWI0VQNSRzNSJMLuYUL+b^LHMJMXZTUR^eb^]lN1IQb^Xf]Z_UH.IKJ.N1VQNSL+NP"VQNSJM[ANPLuYSPMXfxNP;P¿pSN-IQP.H4_QxIKXlPim ³ J.IQVKXlP4YUL1IKRyH.INP"YSPMH.IKJ.qSXlj^IQP J.NSXZYURWNPNVKb^[?YULNP [ANSRyH.NP

L4, L5

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jWY'[ANPMXl­ipSNj^N[^]ZYURWIKH4Yzm± XfRWPMH4YU^Xf]fXljWYSj^I=j^NPo[ANSRyH.NP

L1, L2

IL3

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MWNH.IKL[AN1j^I0x;XljWYj^I

b^L¤NS2MIKH.NRWNP$[aNSRH.NPs]ZYUeJ4YUR^eIiYURWNP-XfRWPMH4_ixIKXlP&rLuYUXlNSJ-yb?YURH.NL1IKRWNSJ-rOY6LuYSP4Ymmk¹ºPMH.NjzXfu[ANSJ

I~^IKL[^]lN^7ybWI'r-LuYUXlP\YSVKXf]FLuYURH.IKJsb^L£VQNSJM[aNkRWNk[ANSRyH.NL1

VQNSJMJ.IK[ANSRWj^IKRyH.Iuv cIKJMJ4Y6bWI-RWNkL1IQPML1N[ANSRyH.NVQNSJMJ.IK[ANSRWj^IKRyH.I$Y^[^XfH.IKJim·`IYUH.N^zRWIKR^tyb^L+YSP.H.IKJ.qSXlj^Isr`NSWP.IKJMxSYSj^N'RWNP [ANSRyH.NP ]ZYUeJ4YUR^eIiYURWNPXfRWPMH4_ixIKXlP=j^I ^[^XfH.IKJim N1IKRH4YURyH.N^ARWNuViYSP4NujWY |IKJMJ4YuI~zXlPMH.IKL¼b^L+eJ4YURWj^I&Ryb^L1IKJ.N6j^Is[?YUJMHM«lVKb^]ZYSP`j^I[Aq1bWI`g?ViYUL+H.IKL[ANSJ4YUJMXZYUL1IKRyH.IViYU[^HMb^J4YSjWYSP=RWN1[ANSRyH.N

L1

^eIKJ4YURWj^NOb^LIKRWqSL1IKRWNuVQNSR^tWIQVKXlj^NkVQNSL1N eIKeIKRWPMt^XfRWI © ÀNSb^]fH.NSR ¶A@z¶ ¬m

FXfeb^J4Y @ µ ³ NPMXl­ipSNjWYSP`[^JMXfRWVKXf[?YUXlP0J.IQP4P.NSR?TURWVKXZYSPsP.IQVKb^]ZYUJ.IQP`RWN6VKXfRyHMb^J4pSNj^I'YSP.H.IKJ.qSXlj^IQP`[?YUJ4Ye = 0.1

mG\POR ^L1IKJ.NPkYUiIKL J.IIKJ.KRWVKXZY®_SPkP4IKeb^XfRyH.IQPkJ.IQP.P.NSR?TURWVKXZYSPiµr

ν5

rν6

rν16

mG\PO[aNSRH.NPJ.IK[^J.IQP.IKRH4YUL¼NPYSP.H.IKJ.qSXlj^IQPNSWP.IKJMxYSj^NPYUH.r

e = 0.2m

¶

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N−1VQNSJM[aNPwLuYSP.XfxNPiI´xSYUL1NP3VQNSRWPMXlj^IKJ4YUJoP4q

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·`IgWR^XfRWj^NuYSP"xSYUJMXZ_ixIKXlP"J.IKeb^]ZYUJ.IQPh = e sin$ p = sin I sinΩk = e cos$ q = sin I cos Ω

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hk

= e0

sincos

(g0t+$0) +

N∑

j=2

Aj

g0 − gj

sincos

(gjt+ αj)

© ¶ ¬

pq

= sin I0

sincos

(s0t+ Ω0) +

N∑

j=2

Bj

s0 − sj

sincos

(sjt+ βj)

© ¶AB ¬NSRWj^I

e0, $0, I0,Ω0

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gi, si

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Aj, αj , Bj, βj

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g0 = d$dt

Is0 = dΩ

dt

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g0 ' gj

NSbs0 ' sj

[?YUJ4Y®YU]feb^Lj

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¶AB

V4t?YULuYSjWYSPν5, ν6

Iν16

J.IQPM[AIQVKHMXfxSYUL1IKRH.IWXlPMH.Nrν5 ≡ g0 ' g5 = d$J

dt

ν6 ≡ g0 ' g6 = d$S

dt

ν16 ≡ s0 ' s6 = dΩJS

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a, iVQNSRWPMXlj^IKJ4YURWj^N

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