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1. F_lh^djmlbevguodhe_[ZgbcgZmgbnbeyjghfih^\_k_
DZdijb[hj^eygZmqguobkke_^h\Zgbcmgbnbeyj[ue\\_^_g\ijZd
lbdm wdki_jbf_glZ Dmehghf \ ] Z \ ijZdlbdm wdki_jbf_glZevgh]h
hij_^_e_gbyfhf_glh\bg_jpbbl_e–\gZqZe_XIX\_dZImZkkhghfb\gZ
klhys__ \j_fy j_dhf_g^m_lky \k_fb jmdh\h^kl\Zfb ih l_hj_lbq_kdhc b
ijbdeZ^ghc f_oZgbd_ >Zgguc f_lh^ y\ey_lky ^hklZlhqgh lhqguf bkdh
fZy\_ebqbgZijbgZ^e_`Zs_clsZl_evghklbhiulZhij_^_ey_lkykhrb[
dhc\g_kdhevdhkhluoijhp_glZ
Mgbnbeyj khklhbl ba mijm]h]h
kl_j`gyijh\hehdb \_jogbc dhg_p dh
lhjh]h `_kldh aZdj_ie_gjbk3 D gb`
g_fmdhgpmkl_j`gyijbdj_ie_ghaZ`bf
gh_ mkljhckl\hih^\_k gZ dhlhjh_ ih
f_rZxl bkiulm_fh_ l_eh < g_dhlhjuo
kemqZyo bkiulm_fh_ l_eh ijbdj_ieyxl
g_ihkj_^kl\_ggh d mijm]hfm kl_j`gx
>ey bkdexq_gby [hdh\uo dhe_[Zgbc g_
h[oh^bfhqlh[uhkvkl_j`gykh\iZ^ZeZk
hkvxbg_jpbb\k_ckbkl_fu
Ijb ih\hjhl_ ijb[hjZ \f_kl_ k l_
ehf\hdjm]hkbgZg_dhlhjucm]hekbkl_
fZ gZqbgZ_l kh\_jrZlv djmlbevgu_ dhe_[Zgby ?keb ij_g_[j_qv \\b^m
fZehklb
,
khijhlb\e_gb_f \ha^moZ b fhf_glhf bg_jpbb kl_j`gy ^bnn_
j_gpbZevgh_mjZ\g_gb_^\b`_gbyijbf_l\b^
0M
dt
d
JJ
2
2
=+
ϕ
+
)(
i
, (7)
]^_J –fhf_glbg_jpbbbkiulm_fh]hl_eZ
J
i
–fhf_glbg_jpbbih^\_kZ
F –\hkklZgZ\eb\Zxsbc fhf_gl h[mkeh\e_gguc mijm]hklvx fZ-
l_jbZeZkl_j`gy
Ijbg_[hevrhfm]e_aZdjmqb\Zgbyϕfh`ghijbgylvqlhwlhlfhf_gl
ijhihjpbhgZe_gm]emaZdjmqb\Zgby
F = k ϕ, (8)
]^_k –dhwnnbpb_gl`_kldhklbih^\_kZ
Jbk
1. F_lh^djmlbevguodhe_[ZgbcgZmgbnbeyjghfih^\_k_ DZd ijb[hj ^ey gZmqguo bkke_^h\Zgbc mgbnbeyj [ue \\_^_g \ ijZd lbdm wdki_jbf_glZ Dmehghf \ ] Z \ ijZdlbdm wdki_jbf_glZevgh]h hij_^_e_gbyfhf_glh\bg_jpbbl_e–\gZqZe_XIX\_dZImZkkhghfb\gZ klhys__ \j_fy j_dhf_g^m_lky \k_fb jmdh\h^kl\Zfb ih l_hj_lbq_kdhc b ijbdeZ^ghc f_oZgbd_ >Zgguc f_lh^ y\ey_lky ^hklZlhqgh lhqguf bkdh fZy\_ebqbgZijbgZ^e_`Zs_clsZl_evghklbhiulZhij_^_ey_lkykhrb[ dhc\g_kdhevdhkhluoijhp_glZ Mgbnbeyj khklhbl ba mijm]h]h kl_j`gy ijh\hehdb \_jogbc dhg_p dh lhjh]h `_kldh aZdj_ie_g jbk3 D gb` g_fmdhgpmkl_j`gyijbdj_ie_ghaZ`bf gh_ mkljhckl\h ih^\_k gZ dhlhjh_ ih f_rZxl bkiulm_fh_ l_eh < g_dhlhjuo kemqZyo bkiulm_fh_ l_eh ijbdj_ieyxl g_ihkj_^kl\_ggh d mijm]hfm kl_j`gx >ey bkdexq_gby [hdh\uo dhe_[Zgbc g_ h[oh^bfhqlh[uhkvkl_j`gykh\iZ^ZeZk hkvxbg_jpbb\k_ckbkl_fu Ijb ih\hjhl_ ijb[hjZ \f_kl_ k l_ Jbk ehf\hdjm]hkbgZg_dhlhjucm]hekbkl_ fZ gZqbgZ_l kh\_jrZlv djmlbevgu_ dhe_[Zgby ?keb ij_g_[j_qv \\b^m fZehklb, khijhlb\e_gb_f \ha^moZ b fhf_glhf bg_jpbb kl_j`gy ^bnn_ j_gpbZevgh_mjZ\g_gb_^\b`_gbyijbf_l\b^ d 2ϕ (J + Ji ) + M =0, (7) 2 dt ]^_J –fhf_glbg_jpbbbkiulm_fh]hl_eZ Ji –fhf_glbg_jpbbih^\_kZ F – \hkklZgZ\eb\Zxsbc fhf_gl h[mkeh\e_gguc mijm]hklvx fZ- l_jbZeZkl_j`gy Ijbg_[hevrhfm]e_aZdjmqb\Zgbyϕfh`ghijbgylvqlhwlhlfhf_gl ijhihjpbhgZe_gm]emaZdjmqb\Zgby F = k ϕ, (8) ]^_k –dhwnnbpb_gl`_kldhklbih^\_kZ 6
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