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9
m
–
fZkkZl_eZ
g –mkdhj_gb_k\h[h^gh]hiZ^_gby
?kebm]heϕ
g_[hevrhclhfh`ghkqblZlvsin ϕ =
ϕ, lh]^Z
ϕ⋅−=
ϕ
mgc
dt
d
J
2
2
. (15)
J_rZy^Zggh_^bnn_j_gpbZevgh_mjZ\g_gb_hij_^_ey_fi_jbh^dhe_
[Zgbynbabq_kdh]hfZylgbdZ
PJF
-
L π= . (16)
?keb iheh`_gb_p_gljZfZkkohjhrhba
\_klghZ\_kmgbaf_j_ggZlhqguo\_kZolh
^ey hij_^_e_gby fhf_glZ bg_jpbb l_eh ih^
\_rb\Zxl gZ g_ih^\b`ghc ]hjbahglZevghc
hkblZdqlh[uhghfh]ehdZqZlvkyih^^_ckl
\b_f kh[kl\_ggh]h \_kZ [_a hkh[h]h lj_gby
Ih\_jgm\ l_eh \hdjm] k\h_c hkb ih^\_kZ
gZ 5 – 8
h
\u\h^yl _]h ba jZ\gh\_kby b
hlimkdZxl [_a gZqZevghc kdhjhklb Baf_jb\
k_dmg^hf_jhf ijh^he`bl_evghklv – dhe_[Zgbc \uqbkeyxl i_jbh^
h^gh]hihegh]hfZeh]hdhe_[ZgbyZ aZl_ffhf_gl bg_jpbb hlghkbl_evgh
hkbih^\_kZihnhjfme_
mgc
T
J
2
2
4π
=
. (17)
Hq_gv qZklh fhf_gl bg_jpbb bkiulm_fh]h l_eZ \uqbkeyxl hlgh-
kbl_evghp_gljZevghchkbiZjZee_evghchkbih^\_kZLh]^Z,kh]eZkghl_h
j_f_ h iZjZee_evguo hkyo, fhf_gl bg_jpbb hlghkbl_evgh hkb ijh-
oh^ys_cq_j_ap_gljfZkk\ujZ`Z_lkynhjfmehc
2
2
4
2
mcmgc
T
J
o
−
π
=
. (18)
3. F_lh^lj_ogblygh]hih^\_kZ
IjbjZaebqguowdki_jbf_glZo\nbabd_bl_ogbd_ijbf_gyxlih^\_
rb\ZgbygZ]b[dbogblyobebe_glZo\_jogb_dhgpudhlhjuoaZdj_ie_gu
Jbk 4
m –fZkkZl_eZ
g –mkdhj_gb_k\h[h^gh]hiZ^_gby
?kebm]heϕ g_[hevrhclhfh`ghkqblZlvsin ϕ = ϕ, lh]^Z
dϕ 2
J = − mgc ⋅ ϕ . (15)
dt 2
J_rZy^Zggh_^bnn_j_gpbZevgh_mjZ\g_gb_hij_^_ey_fi_jbh^dhe_
[Zgbynbabq_kdh]hfZylgbdZ
-
L = π . (16)
PJF
?keb iheh`_gb_ p_gljZ fZkk ohjhrh ba
\_klghZ\_k mg baf_j_ggZlhqguo\_kZolh
^ey hij_^_e_gby fhf_glZ bg_jpbb l_eh ih^
\_rb\Zxl gZ g_ih^\b`ghc ]hjbahglZevghc
hkblZdqlh[uhghfh]ehdZqZlvkyih^^_ckl
\b_f kh[kl\_ggh]h \_kZ [_a hkh[h]h lj_gby
Ih\_jgm\ l_eh \hdjm] k\h_c hkb ih^\_kZ
gZ 5 – 8 h \u\h^yl _]h ba jZ\gh\_kby b Jbk 4
hlimkdZxl [_a gZqZevghc kdhjhklb Baf_jb\
k_dmg^hf_jhf ijh^he`bl_evghklv – dhe_[Zgbc \uqbkeyxl i_jbh^
h^gh]h ihegh]h fZeh]h dhe_[Zgby Z aZl_f fhf_gl bg_jpbb hlghkbl_evgh
hkbih^\_kZihnhjfme_
T2
J= mgc . (17)
4π 2
Hq_gv qZklh fhf_gl bg_jpbb bkiulm_fh]h l_eZ \uqbkeyxl hlgh-
kbl_evghp_gljZevghchkbiZjZee_evghchkbih^\_kZLh]^Z,kh]eZkghl_h
j_f_ h iZjZee_evguo hkyo, fhf_gl bg_jpbb hlghkbl_evgh hkb ijh-
oh^ys_cq_j_ap_gljfZkk\ujZ`Z_lkynhjfmehc
T2
Jo = mgc − mc 2 . (18)
4π 2
3. F_lh^lj_ogblygh]hih^\_kZ
IjbjZaebqguowdki_jbf_glZo\nbabd_bl_ogbd_ijbf_gyxlih^\_
rb\ZgbygZ]b[dbogblyobebe_glZo\_jogb_dhgpudhlhjuoaZdj_ie_gu
9
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