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    aa′, Λ

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    ,kΛb′′

    b′ ∆b′

    a′k Λa′

    a′′ Λa′′

    c′ uc′ + Λb

    ′′

    a′ Λa′

    a′′,k Λa′′

    c′ uc′ = 0. (57)

    Ê k_zwh�`cq_zws�`c]�oH[�h�i^a`cb�d:^�[�mG §è�ã�¢|5sΛd

    b′′g�]_b^ak_nrno`c]�y[5d:e

    b′′£*v|d"�d�h

    Λd′

    b′′ Λb′′

    b′,k ub′ + ud

    ,k − ∆d′

    c′k uc′ + Λd

    a′′,k Λa′′

    c′ uc′ = 0. (58)

    Ð d=]�[�hadjh�i%gIh�h�i�dj¥�ep^fh9g�]_b?z{g�^fh|had:epnr^\[�m| §è�ç5¢�t:g�]_t�d:z�[5k*h3b_k�djha[Ëh�i�d=efd:z{gIh`0[5]_^ai_`cq

    Λd′

    b′′ Λb′′

    b′,k + Λd′

    a′′,k Λa′′

    b′ = 0.

    Ð d=v9`czczh�i�k_^9g�epep`w5dgIhGh�i�d"d:}�k%gIh�`0[5]

    ud′

    ,k − ∆d′

    c′kuc′ = 0, (59)

    v9i_`ctiÞ`c^/h�i�d�[5ep`0d:]5hgIh�`0[5]Üd:}�k%gIh�`c[5]ßmx[5e?g�]Ü[5ep`0d:]5had:bÞqH[5`c]�h`c]g�]�[�h�i�d:ermeag�nod¨[�mefd�mxd:efd:]_t�d5 Ð dÛi%g�5d¨h�i�k_^$^ai�[v9]ßh�i%gIh$h�i�dmx[5epn [�m=h�i�d¨[5ep`0d:]5hgIh�`0[5]Üd:}�k%gIh�`0[5]Ü`c^`c]_b�d:q�d:]_b�d:]5hV[�mHh�i�d3ti�[5`{t�d|[�m�g9mx[5k_e'Db_`{nod:]_^a`w[5]%g�z*ef[�hgIh�`c]�meag�nod5 ]X[�h�i�d:e�v|[5epb_^u£h�i�d¦[5ep`0d:]5hgIh�`0[5]d:}�k%gIh�`0[5]`c^9`c]��g�ep`{g�]5h6k_]_b�d:e�h�i�d"h�eag�]_^fmx[5epn/gIh�`0[5]?5ep[5k_q

    SO(1.3)

    ]gG^a`cnr`cz{gLe�n/g�]_]�d:e�£v|d�t:g�]"q_ep[:5dlh�i�d�`c]5�g�ep`{g�]_t�dlk_]_b�d:e�h�eag�]_^fmx[5epn/gIh�`0[5]_^SO(1.3)[�m�h�i�d"n/gIhep`;ª$d:}�k%gIh�`0[5]_^¦ x×&ç5¢

    d [v^ai�[v±h�i%gIh|h�i�dq_ef[5q�[5^pd:b/d:}�k%gIh�`0[5]r[�m[5ep`0d:]5hgIh�`0[5]/b�[�d:^|]�[�h�t�[5]5hg�`c]/b_`cefd:th`c]*mx[5epn/gIh�`0[5]g��[5k*h3h�i�d¦no[�h�`0[5]$[�mVgXq%g�eBh�`ctuz0drÔ�g�^9gËv9i�[5z0d�ÔL£_`©�d5c£*`wh9b�[�d:^3]�[�h3dª�d:eBh

  • g�]5s

  • gXq%g�eBh�`ctuz0dr m`c^|h�i�d¦q%g�eBh�`ctuzcd5^�n/g�^a^¢

    m dui

    ds+ m ∆ijk u

    j uk = 0,

    m dua

    ds− m ∆abk ub uk = 0.

    (63)

    Gi�d~^pd�h

  • ¥%d:zcb� ]m©g�th:£g�]5sdª*had:ep]%g�zQ¥%d:zcb¨`c^6ti%g�eag�thad:ep`0ud:b5s�g/^pd�h6[�m�t�[5nrq�[5]�d:]5h�^3[�mVh�i�dZ|i_ep`c^fha[�¯d:z�^fs�ny�[5zc^

    Γijkg�]_bË[�m�h�i�dGt�[5nrq�[5]�d:]5h�^�[�m%h�i�d9ÍG`ctutu`*ef[�hgIh�`0[5]yt�[�d�ý/tu`0du]5h�^

    T abkÕ�èÙ®5ÑGd:efd6gq_k_efd:zwsËha[5ep^a`0[5]Xdª*had:ep]%g�z*¥%d:zcbX`c^Vq�[5^a^a`c_z0d�h�i%gIh�efd�mxd:ep^ha[=h�i�d9`c]�d:eBh�`{g�z

    meag�nod[�m�efd�mxd:efd:]_t�dΓijk = 0,

    T abk 6= 0,(67)

    v9i_`ctigI¯d:th�^|h�i�d=mx[5k_e''b_`cnod:]_^a`0[5]%g�zH[5ep`0d:]5hgIh�`0[5][�mDh�i�d"[5ep`0d:]5had:bq�[5`c]5h:Ð d|]�[�had�h�i%gIh�h�i�d|d:}�k%gIh�`0[5]¦[�m�no[�h�`0[5]g�]_b¦h�i�d\[5ep`0d:]5hgIh�`0[5]¦d:}�k%gIh�`0[5]b�d�had:epnr`c]�d5£

    �d:]�d:eag�zczws$^aq�dg�¬�`c]�%£�h�i�d"t�[5nrq�[5]�d:]5h�^\[�m�[5]�d=}�k%g�]5h�`wh's/h�i%gIh9`c^�t�[5]_t�d:ep]�d:b/v9`wh�i?h�i�d[5ep`0d:]5had:b

  • Gi�d:efd�mx[5efd5£Dh�i�d^pd�hX[�m6d:}�k%gIh�`0[5]_^ §÷�å5¢`c]8g�]8`c]�d:eBh�`{g�z�meag�nod/v9`wh�i�[5k*hodª*had:ep]%g�z¥%d:zcb_^GHd:t�[5nodu^

    mdui

    ds= 0,

    mdua

    ds= 0.

    (71)

    mlv|dËh�i�d:]Ûq%g�^a^j[:5d:e9mef[5ng�]Û`c]�d:eBh�`{g�z©£H`©�d5c£H]�[5]_ef[�hgIh�`c]�?meag�nod5£_ha[g/ef[�hgIh�`{]�mx[5k_e''b_`cnod:]_^a`0[5]%g�z�me�g�nod[�mefd�mxd:efd:]_t�d5£�mx[5epnyk_z{g? §è�á�¢�^a`cnrq_zc`;¥%d:^\ha[

    ∆a′

    b′k = Λa′

    a ∆abk Λ

    bb′ − Λa

    a Λab′,k = −Λa

    a Λab′,k. (72)

    Gi�d~[5ep`0d:]5hgIh�`0[5]Þd:}�k%gIh�`0[5]Þmx[5e?gq%g�eBh�`ctuz0d~efd:z{gIh�`w5d¨ha[ßg@meag�nodÛ`c]5�[5zw5d:b±`c]gmx[5k_e''b_`cnod:]_^a`0[5]%g�zef[�hgIh�`0[5]t:g�]$h�i�k_^9�dv9ep`whahad:]g�^

    mdua

    ds+ m Λa

    a Λab′,k u

    b′ uk = 0, (73)

    v9i�d:efdΛa

    a ∈ SO(1.3)g�efdn/gIh�ep`ct�d:^G[�mh�eag�]_^a`wh�`0[5]mef[5n g�]`c]�d:eBh�`{g�z� ]�[5]_ef[�hgIh�`c]��¢

    meag�nod|ha[¦gef[�hgIh�`c]�meag�nod�[�mMefd�mxd:efd:]_t�d5 Ð d3]�[�hadGh�i%gIh�`c] §ã�å5¢�g�efd�h�i�d3t�[5nrq�[5]�d:]5h�^[�mVgmx[5k_e'\b_`cnod:]_^a`0[5]%g�zH5d:z0[*tu`wh's/[�mVgXq%g�eBh�`ctuz0d=`c]?h�i�d¦`c]_`wh�`{g�z�]�[5]_ef[�hgIh�`c]�Xmeag�nod5Z\[5]_^a`cb�d:e�g�]Ë`c]_`wh�`{g�z*`c]�d:eBh�`{g�z5meag�nod\[�mHefd�mxd:efd:]_t�d5£I^ak_tih�i%gIhV`c]Ë`whlg6q%g�eBh�`ctuz0d\v\[5k_z{b

    ]�[�h9no[5d=efd:z{gIh�`w5dha[r`wh6g�]_b?v|[5k_zcbefd:^a`cb�d¦gIh9`wh�^�[5ep`05`c]�_Gi�k_^u£

    uk =

    1000

    , (74)

    g�]_b §ã�å5¢|efd:b_k_t�d:^�ha[/gX^a`cnrq_z0due\dª_q_efd:^a^a`0[5]

    mdua

    ds= −m Λa′a Λab′,0 ub

    . (75)

    Gi_`c^yd:}�k%gIh�`0[5]`c^v9ep`whphad:]-`c]h�i�dz{gus5d:eËt�[�[5epb_`c]%gIhad:^Ë[�m3h�i�d?meag�nod?q�d:eBmx[5epnr`c]�gXmx[5k_e''b_`cnod:]_^a`0[5]%g�zQef[�hgIh�`0[5]¨`c]h�i�dy`c]_`wh�`{g�z`c]�d:eBh�`{g�z�meag�nod"[�mlefd�mxd:efd:]_t�d5%Gi�dyep`05i�h^a`cb�dX[�m�h�i�dXd:}�k%gIh�`0[5]~t�[5]5hg�`c]_^"g?t�[5k_]5had:epq%g�eBh"[�mlmx[5eat�dydª_q_efd:^a^pd:bÛ`c]Ûhad:epnr^j[�m�h�i�dz0[*t:g�z�z{gus�d:eGt�[�[5epb_`c]%gIhad:^

    F a′

    = −m Λa′a Λab′,0 ub′

    = m Λa′

    a,0 Λab′ u

    b′. (76)

    �n�

  • Gi_`c^3^ak���d:^fh�^|h�i%gIh9^p[5zw�`c]� §ã�è�¢£*v|d=v3`{zcz[5*hg�`c]?h�i�d"q%g�eBh`ctuz0d=q%gIh�i`c]?had:epnr^|[�mh�i�djz0[*t:g�z�z{gus5d:elt�[*[5epb_`c]%gIhad:^u&[Ët�[5nrq%g�efd3v9`wh�irdª_qHd:ep`cnodu]5hg�z%b%gIhg"v|d9v9`czcz�i%gu5d3h�[q_ef[I�Bd:th"h�i_`c^¦q%gIh�i[5]

  • ] ©ç5á�¢£Λa

    a

    g�efd?h�i�dn/gIh�ep`ct�d:^[�mjef[�hgIh�`0[5]-`c]8gÛq_^pd:k_b�[LfÏ�k_tuzc`cb_`{g�]^aq%g�t�d5£Dv9i_`ctit�[5epefd:^aq�[5]_bha[

  • Ð dv9`czcz3g�^a^p[*tu`{gIhadh�i�d¥�ep^fh$t:g�^pdv9`wh�i-hi�d¨]�[5]_`c]�d:eBh�`{g�z�meag�nod[�mjh�i�d¥�ep^fh$¬�`c]_b�«g�]_b�h�i�do^pd:t�[5]_b�£Hha[¨g$]�[5]_`c]�d:eBh�`g�zDmeag�nodË[�m�h�i�do^pd:t�[5]_bÛ¬�`c]_b� Ð dyv9`czcz^ai�[vóh�i%gIhmx[5ept�d:^9g�th�`c]�o[5]¨gXq%g�eBh�`ctuz0d¦`c]?hi�d=h'v|[omeag�nod:^G`c]}�k�d:^fh�`0[5]g�efd¦b_`w¯d:efd:]5h:£Hg�]_bv9`czcz`c]5had:epq_efd�hGh�i�d¦efd:^ak_zwh�^�[5*hg�`c]�d:b�*k__^fh�`wh�k*h�`c]�ß ©ç�÷5¢¦g�]_b ©ç5ã�¢¦`c]5ha[- ©ç5á�¢g�]_bk_^a`c]�¨h�i�d$mx[5zcz0[v9`c]��q_ef[5q�d:eBh�`0d:^[�m

    }�k%g�b_eagIh�`ct"n/gIh�ep`ct�d:^

    (A B)−1 = B−1 A−1, A−1 A = I,v|d�v9`czcz�b�d:ep`w5dmx[5epnyk_z{g�^mx[5e

    P i′

    j′mx[5eh�i�dVh®v|[9t:g�^pd:^u£dª_q_efd:^a^pd:b=`c]jhad:epnr^M[�m�q%g�eag�nod�had:ep^

    [�m�q_k_efd:zws?^aq%gIh�`{g�zQef[�hgIh�`0[5]g�]_b[�m�ef[�hgIh�`0[5]`c]gXq_^pd:k_b�[Lf�k_tuzc`cb�dg�]q_z{g�]�d

    Pi′ (RL)j′ = e

    i′

    a

    (

    Λa′′

    a

    )−1Λa

    ′′

    b,0 ebj′ =

    = δi′

    a

    (

    Ra′′

    a′ La′

    a

    )−1 (Ra

    ′′

    a′ La′

    b

    )

    ,0δbj′ =

    = δi′

    a

    [

    (

    La′

    a

    )−1 (Ra

    ′′

    a′

    )−1Ra

    ′′

    a′,0 La′

    b +(

    La′

    a

    )−1La

    b,0

    ]

    δbj′,

    (88)

    mx[5e3gX]�[5]_`c]�d:eBh�`{g�zmeag�nod[�m�h�i�d=¥�ep^fh9¬�`c]_b�£�g�]_b

    Pi′ (LR)j′ = δ

    i′

    a

    [

    (

    Ra′

    a

    )−1 (La

    ′′

    a′

    )−1La

    ′′

    a′,0 Ra′

    b +(

    Ra′

    a

    )−1Ra

    b,0

    ]

    δbj′, (89)

    mx[5e3gX]�[5]_`c]�d:eBh�`{g�zmeag�nod[�m�h�i�d¦^pd:t�[5]_b¬�`c]_b�o ¶¶#´·�Á9À_¹QÉfÈVÇ*»qp3»p¼&Ç*»pųÈVÂjÂ6À_·(r9»f¸ëÈ�Ç*»B·\Áy¶Gi_`c^^pd:th�`0[5]±g�b_b_efd:^a^pd:^/h�i�d

  • v9i�d:efdϕ = ϕ(t)

    `c^�h�i�dg�]�5z0d=[�mef[�hgIh�`0[5]¨g��[5k*hGh�i�dzgª_`c^u

    Gi�d�n/gIh�ep`;ª-[�m=q_^pd:k_b�[LfÏ�k_tuzc`cb�dg�]ßef[�hgIh�`0[5]�£�ha[v9`wh�i_`c]o(β)

    £l`c^odª_q_efd:^a^pd:bÜg�^mx[5zcz0[v9^uÓ

    L =

    1 −βxγ −βyγ 0−βxγ 1 0 0−βyγ 0 1 0

    0 0 0 1

    , (92)

    v9i�d:efdβx = βx(t), βy = βy(t), β

    2 = β2(t) = β2x(t) + β2y(t).Ð `wh�i�[5k*hGz0[5^a^�[�mQ�d:]�d:eag�zc`wh's&£�v|d9v9`czczg�^a^ak_nod3h�i%gIh3gIhGg"h�`cnod

    th�i�dj[5ep`05`c]_^�[�m�h�i�d

    ]�[5]_`c]�d:eBh�`{g�zQg�]_b`{]�d:eBh�`{g�zHmeag�nod:^�t�[5`c]_tu`cb�d5s `cefd:th3t�[5nrq_k*hgIh�`0[5]?[�m�h�i�d¦n/gIh�ep`;ª?d:z0d:nXd:]�h^

    P i′

    j′ha[ov3`0h�i_`c]

    o(β)5`w5d:^

    Pi′ (RL)j′ =

    1

    c

    0 −β̇x − βyΩ −β̇y + βxΩ 0−β̇x − βyΩ 0 Ω 0−β̇y + βxΩ −Ω 0 0

    0 0 0 0

    , (93)

    Pi′ (LR)j′ =

    1

    c

    0 −β̇x cos ϕ + β̇y sin ϕ −β̇y cos ϕ − β̇x sin ϕ 0−β̇x cos ϕ + β̇y sin ϕ 0 Ω 0−β̇y cos ϕ − β̇x sin ϕ −Ω 0 0

    0 0 0 0

    .

    (94)ê_ef[5n §ã�à5¢£�v|d6v9`czcz�v9ep`whadjh�i�dd:}�k%gIh�`0[5]/[�m�no[�h�`0[5]rmx[5e|gËq%g�eBh�`ctuz0d9�h�i�d[5ep`05`c]Ý[�mVg�]`c]�d:eBh�`{g�zHmeag�nod=[�mefd�mxd:efd:]_t�dj�`c]gX]�[5]_`c]�d:eBh�`{g�zmeag�nod[�m�efd�mxd:efd:]_t�d5Ó

    γ′

    c

    dui′

    dt= P i

    j′ uj′, (95)

    [5eG`c]n/gIh�ep`;ª$mx[5epnÓ

    γ′

    c

    d

    dt

    γ′

    γ′B′xγ′B′yγ′B′z

    = P i′

    j′

    γ′

    γ′B′xγ′B′y

    0

    . (96)

    h�`c^ha[¦�d3efd:nod:nË�d:efd:bh�i%gIhVh�i�d�5d:z0[*tu`wh'sV

    ′ [�mMq�[5`c]5h@efd:z{gIh�`w5d|ha[g]�[5]_`c]�d:eBh�`{g�zmeag�nodl[�m%efd�mxd:efd:]_t�d�`c^Qd:}�k%g�z&`c]¦n/gL5]_`wh�k_b�d|g�]_b"[5q_q�[5^a`0had�`c]b_`cefd:th�`0[5]ha[9h�i�dl5d:z0[*tu`wh's

    � Ú

  • v[�mDh�i�d"[5ep`05`c][�mDh�i�d¦]�[5]_`c]�d:eBh�`{g�zmeag�nod"efd:z{gIh�`w�dha[Xh�i�d¦`c]�d:eBh�`{g�zmeag�nod5Ó

    V ′α = −vα,B′α = V ′α/c = −βα.

    (97)

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    γ′ =1

    1 − B′2=

    1√

    1 − β2= γ. (98)

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    `c]og]�[5]_`c]�d:eBh�`{g�z�meag�nod|[�mHh�i�d�¥�ep^fhl¬�`c]_bX`c]Ëh�i�d3]�[5]_efd:z{gIh�`w�`c^fh�`ct|t:g�^pd|ha[v9`wh�i_`c]o(β)

    Ï�}�k%gIh�`0[5]mx[5ei′ = 0

    Ódγ′

    dt= −β̇xB′x − β̇yB′y. (99)

    Ê k_zwh�`cq_zws�`c]�Ë�[�h�i?^a`cb�d6[�m�h�i_`c^\d:}�k%gIh�`0[5]/5smc2

    g�]_b$t�[5]_^a`cb�d:ep`c]�¦h�i%gIhE = γ′mc2`c^9h�i�dyt�[5nrq_z0d�had¦d:]�d:ef�s[�mVh�i�dy�[*b*s&£�v|dv9`czcz[5*hg�`c]¨h�i�dËd:}�k%gIh�`0[5]mx[5e3h�i�dXq�[v\d:e

    [�m�h�i�d"`{]�d:eBh�`{g�zHmx[5ept�d:^uÓdE

    dt= −mwxV ′x − mwyV ′y, (100)

    v9i�d:efdwα = β̇αc

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    i′ = 1Ó

    dB′xdt

    = −β̇x − βyΩ + B′yΩ = −β̇x + 2B′yΩ. (101)Ï�}�k%gIh�`0[5]mx[5e

    i′ = 2Ó

    dB′ydt

    = −β̇y + βxΩ − B′xΩ = −β̇y − 2B′xΩ. (102)Ï�}�k%gIh�`0[5]mx[5e

    i′ = 3Ó

    dB′zdt

    = 0. (103)

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  • Ð d?i%gu5d?k_^pd:bi�d:efdβα = −B′α

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    mdV ′xdt

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    mdV ′ydt

    = −mwy − 2mV ′xΩ, (105)

    mdV ′zdt

    = 0. (106)

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    mdV′

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