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ɩɨɥɟɡɧɨɝɨ ɢɫɩɨɥɶɡɨɜɚɧɢɹ ɜɵɞɟɥɹɸɳɟɝɨɫɹ ɬɟɩɥɚ ɗɮɮɟɤɬɢɜɧɨɟ ɢɫɩɨɥɶɡɨɜɚɧɢɟ
:ɜɨɦɧɨɝɨɦɨɩɪɟɞɟɥɹɟɬɷɧɟɪɝɟɬɢɱɟɫɤɭɸɷɮɮɟɤɬɢɜɧɨɫɬɶȾɋɉɌɟɩɥɨɷɤɡɨɬɟɪ
ɦɢɱɟɫɤɢɯɪɟɚɤɰɢɣɨɩɪɟɞɟɥɹɟɦɩɨɮɨɪɦɭɥɟ
:
Pɷɥǻ+ɷɥ
ɝɞɟPɷɥ±ɦɚɫɫɚɷɥɟɦɟɧɬɚɨɤɢɫɥɹɸɳɟɝɨɫɹɡɚɩɟɪɢɨɞɤɝ
ǻ+ɷɥ±ɬɟɩɥɨɜɨɣɷɮɮɟɤɬɪɟɚɤɰɢɢɨɤɢɫɥɟɧɢɹɷɥɟɦɟɧɬɚɆȾɠɤɝ
ɂɫɯɨɞɧɵɦɢɞɚɧɧɵɦɢɞɥɹɪɚɫɱɟɬɚ ɬɟɩɥɨɜɨɝɨɷɮɮɟɤɬɚ ɯɢɦɢɱɟɫɤɨɣɪɟɚɤɰɢɢ
ɦɨɝɭɬ ɫɥɭɠɢɬɶ ɬɟɩɥɨɬɵ ɨɛɪɚɡɨɜɚɧɢɹ ɜɟɳɟɫɬɜ ɭɱɚɫɬɜɭɸɳɢɯ ɜ ɪɟɚɤɰɢɢ Ⱦɥɹ
ɪɚɫɱɟɬɚɬɟɩɥɨɜɨɝɨɷɮɮɟɤɬɚɪɟɚɤɰɢɢɩɪɢɥɸɛɨɣɬɟɦɩɟɪɚɬɭɪɟɌɢɫɩɨɥɶɡɭɸɬɫɥɟɞɭ
ɸɳɟɟɜɵɪɚɠɟɧɢɟ
ɝɞɟ
Ɍ
R
ɇΔ
±ɬɟɩɥɨɜɨɣɷɮɮɟɤɬɪɟɚɤɰɢɢɜɫɬɚɧɞɚɪɬɧɵɯɭɫɥɨɜɢɹɯɤȾɠɦɨɥɶ
ɋΔ
±ɬɟɩɥɨɟɦɤɨɫɬɶȾɠɄāɦɨɥɶ
ɪ
Ɍ
Δ+Δ=Δ
³
GɌɋɇɇ
ɪ
ȿɫɥɢɜɧɭɬɪɢɪɚɫɫɦɚɬɪɢɜɚɟɦɨɝɨɢɧɬɟɪɜɚɥɚɬɟɦɩɟɪɚɬɭɪ ɨɞɢɧɢɥɢɧɟɫɤɨɥɶɤɨ
ɭɱɚɫɬɧɢɤɨɜɪɟɚɤɰɢɢɢɫɩɵɬɵɜɚɸɬɮɚɡɨɜɵɟɩɪɟɜɪɚɳɟɧɢɹɬɟɦɩɟɪɚɬɭɪɧɵɣɢɧɬɟɪ
ɜɚɥɪɚɡɛɢɜɚɸɬɧɚɧɟɫɤɨɥɶɤɨɱɚɫɬɟɣ
R
Ɍ
ȼɤɚɠɞɭɸɚɥɝɟɛɪɚɢɱɟɫɤɭɸɫɭɦɦɭ
R
Ɍ
ɩɪɟɜɪ
S
ɋΔ
ɋΔ
ɢ
S
Ɍ
Δ+Δ⋅ν±Δ+Δ=Δ
G7ɋɇG7ɋɇɇ
Sɩɪɟɜɪ
³³
Ɍ
ɩɪɟɜɪ
ɜɯɨɞɹɬɬɟɩɥɨɟɦɤɨɫɬɢɬɟɯɮɚɡ
S
ɤɨɬɨɪɵɟ ɭɫɬɨɣɱɢɜɵ ɜ ɞɚɧɧɨɦ ɢɧɬɟɪɜɚɥɟ ɬɟɦɩɟɪɚɬɭɪ Ɍɟɩɥɨɬɚ ɩɪɟɜɪɚɳɟɧɢɹ
ǻɇ
ɛɟɪɟɬɫɹɫɨɫɜɨɢɦɢɥɢɩɪɨɬɢɜɨɩɨɥɨɠɧɵɦɡɧɚɤɨɦɜɡɚɜɢɫɢɦɨɫɬɢɨɬɩɪɢɧɚɞ
ɩɪɟɜɪ
ɥɟɠɧɨɫɬɢɤɩɪɨɞɭɤɬɭɪɟɚɤɰɢɢɢɥɢɢɫɯɨɞɧɨɦɭɜɟɳɟɫɬɜɭɌɚɤɠɟɨɩɪɟɞɟɥɟɧɢɟɬɟɩ
ɥɨɜɨɝɨɷɮɮɟɤɬɚɪɟɚɤɰɢɢɜɨɡɦɨɠɧɨɢɡɜɵɪɚɠɟɧɢɹ
Ɍ
Ɍ
67+*
Δ⋅−Δ=Δ
Ɍ
Ɍɟɩɥɨɷɤɡɨɬɟɪɦɢɱɟɫɤɢɯ ɪɟɚɤɰɢɣɨɩɪɟɞɟɥɹɟɦ ɫɭɦɦɢɪɨɜɚɧɢɟɦɬɟɩɥɨɬ ɪɟɚɤ
ɰɢɣɢɞɭɳɢɯɫɨɬɪɢɰɚɬɟɥɶɧɵɦɷɮɮɟɤɬɨɦ
>6L@^Ɉ`ĺ 6L2 ⋅ ɆȾɠ
>ɆQ@^Ɉ`ĺɆQɈ ⋅ ɆȾɠ
>)H@^2`ĺ)H2 ⋅ ɆȾɠ
>)H@^Ɉ`ĺ)H2 ⋅ ɆȾɠ
^ɋɈ`^Ɉ`ĺ^ɋɈ` ⋅ ɆȾɠ
>ɋ@^Ɉ`ĺ^ɋɈ` ⋅ ɆȾɠ
>Ɋ@^Ɉ`ĺɊɈ ⋅ ɆȾɠ
>ɋU@^Ɉ`ĺɋUɈ ⋅ ɆȾɠ
>6L@)H2ĺ6L2>)H@ ⋅ ɆȾɠ
>0Q@)H2ĺ0Q2>)H@ ⋅ ɆȾɠ
>3@)H2ĺ32>)H@ ⋅ ɆȾɠ
>&U@)H2ĺ&U2>)H@ ⋅ ɆȾɠ
ɋɥɟɞɨɜɚɬɟɥɶɧɨ: ȽȾɠ
ɗɧɟɪɝɢɹɩɨɫɬɭɩɚɸɳɚɹɨɬɝɚɡɨɤɢɫɥɨɪɨɞɧɵɯɝɨɪɟɥɨɤ
τ
31:
⋅⋅=
ɝɞɟ1±ɤɨɥɢɱɟɫɬɜɨɪɚɛɨɬɚɸɳɢɯɝɨɪɟɥɨɤ
Ɋ±ɦɨɳɧɨɫɬɶɝɨɪɟɥɨɤȼɬ
IJ±ɜɪɟɦɹɪɚɛɨɬɵɝɨɪɟɥɨɤɦɢɧ
Ɍɨɝɞɚ ȽȾɠɱɆȼɬ
Ɍɟɩɥɨɲɥɚɤɨɨɛɪɚɡɨɜɚɧɢɹ
6L2ĺ&D26L2 Ⱦɠ
ɋɥɟɞɨɜɚɬɟɥɶɧɨ
:
⋅=⋅⋅⋅=
=⋅+=
⋅=⋅⋅
ȽȾɠ:
Ɍɟɩɥɨɜɵɞɟɥɟɧɧɨɟɜɩɟɱɢɨɬɨɤɢɫɥɟɧɢɹɝɪɚɮɢɬɢɪɨɜɚɧɧɵɯɷɥɟɤɬɪɨɞɨɜɦɨ
ɠɧɨɨɩɪɟɞɟɥɢɬɶɬɨɥɶɤɨɨɪɢɟɧɬɢɪɨɜɨɱɧɨɧɟɫɦɨɬɪɹɧɚɬɨɱɬɨɜɡɜɟɲɢɜɚɧɢɟɦɦɨ
ɠɧɨɬɨɱɧɨɭɫɬɚɧɨɜɢɬɶɦɚɫɫɭɢɡɪɚɫɯɨɞɨɜɚɧɧɵɯɡɚɩɥɚɜɤɭɢɥɢɡɚɩɟɪɢɨɞɷɥɟɤɬ
ɪɨɞɨɜɉɪɢɨɩɪɟɞɟɥɟɧɢɢ:ɫɥɟɞɭɟɬɩɨɦɧɢɬɶɱɬɨɧɟɜɫɹɦɚɫɫɚɢɡɪɚɫɯɨɞɨɜɚɧɧɵɯ
ɷɥɟɤɬɪɨɞɨɜɫɝɨɪɚɟɬɜɩɟɱɢɑɚɫɬɶɭɝɥɟɪɨɞɚɷɥɟɤɬɪɨɞɨɜɨɤɢɫɥɹɟɬɫɹɥɢɲɶɞɨɨɤɢɫɢ
ɚɱɚɫɬɶɭɧɨɫɢɬɫɹɫɝɚɡɚɦɢɜɜɢɞɟɫɚɠɢɭɱɟɫɬɶɤɨɬɨɪɭɸɧɟɜɨɡɦɨɠɧɨɢɡɡɚɩɨɩɭɬ
ɧɨɝɨɭɧɨɫɚɢɡɩɟɱɢɢɞɪɭɝɢɯɦɟɯɚɧɢɱɟɫɤɢɯɤɨɦɩɨɧɟɧɬɨɜȺȾɋɜɟɧɱɚɧɫɤɢɣɪɟ
ɤɨɦɟɧɞɭɟɬɭɱɢɬɵɜɚɬɶ ɜɩɪɢɯɨɞɧɨɣ ɱɚɫɬɢɛɚɥɚɧɫɚ ɥɢɲɶ ɨɬ ɩɨɬɟɧɰɢɚɥɶɧɨ
ɜɨɡɦɨɠɧɨɝɨ:Ɍɨɝɞɚɬɟɩɥɨɨɬɨɤɢɫɥɟɧɢɹɷɥɟɤɬɪɨɞɨɜ
:PT=⋅Δ⋅
ɗɋ
ɝɞɟǻPɷ±ɦɚɫɫɚɢɡɪɚɫɯɨɞɨɜɚɧɧɵɯɷɥɟɤɬɪɨɞɨɜɤɝ
Tɋ±ɬɟɩɥɨɜɨɣɷɮɮɟɤɬɨɤɢɫɥɟɧɢɹɝɪɚɮɢɬɚɆȾɠɤɝ
Ɍɨɝɞɚ
Ɍɟɩɥɨɜɧɨɫɢɦɨɟɦɟɬɚɥɥɨɦɢɲɥɚɤɨɦɨɬɩɪɟɞɵɞɭɳɟɣɩɥɚɜɤɢ
ɝɞɟP
±ɦɚɫɫɚɦɟɬɚɥɥɚɢɲɥɚɤɚɤɝ
ɦɲɥ
=⋅⋅=
:::=+
:P&W=⋅⋅
:P&W=⋅⋅
ɲɥ ɲɥ ɲɥ ɲɥ
ȽȾɠ:
ɦɲɥ
ɦɦɦɦ
: ȽȾɠ=⋅⋅+ ⋅⋅=
[
[
Ɍɨɝɞɚ
Ɍɟɩɥɨɜɧɨɫɢɦɨɟɜɨɡɞɭɯɨɦɢɝɚɡɨɨɛɪɚɡɧɵɦɤɢɫɥɨɪɨɞɨɦ
7F9:
⋅⋅=
ɝɞɟ9±ɨɛɴɟɦɜɨɡɞɭɯɚɢɥɢɤɢɫɥɨɪɨɞɚɦ
ɫ±ɬɟɩɥɨɟɦɤɨɫɬɶɜɨɡɞɭɯɚɢɥɢɤɢɫɥɨɪɨɞɚɤȾɠɦāɄ
Ɍ±ɬɟɦɩɟɪɚɬɭɪɚɜɨɡɞɭɯɚɢɥɢɤɢɫɥɨɪɨɞɚɄ
Ɍɨɝɞɚ
: Ⱦɠ=⋅⋅=⋅
ɜ
: Ⱦɠ=⋅⋅=⋅
ɤ
: : : ȽȾɠ=+= + ⋅=
ɜɤ
()
ɋɬɚɬɶɢɪɚɫɯɨɞɚɬɟɩɥɨɬɵ
ɉɨɥɟɡɧɵɣɪɚɫɯɨɞɷɧɟɪɝɢɢ
: : : =+
ɦɲɥ
ɝɞɟ:ɦ±ɮɢɡɢɱɟɫɤɨɟɬɟɩɥɨɫɬɚɥɢȾɠ
:ɲɥ±ɮɢɡɢɱɟɫɤɨɟɬɟɩɥɨɲɥɚɤɚȾɠ
ɝɞɟɋ
ɲ
:Pɋ>W W@TPɋ>W W@T=⋅−++⋅−+
ɦɩɥɦ ɩɥɦ
ɲ ɲ ɧ ɲ ɲ ɧ
ɠ
: P P ɋ W W P & W W =+⋅⋅−+⋅⋅−
ɦɦɜɵɩɩɥɠɱɱɜɵɩɠɱ
ɲ ɲ
:::=+
ɦɦɦ
±ɫɪɟɞɧɹɹɬɟɩɥɨɟɦɤɨɫɬɶɦɟɬɚɥɥɚɜɢɧɬɟɪɜɚɥɟɨɬWɧ±ɬɟɦɩɟɪɚɬɭɪɵɡɚɝɪɭ
ɠɚɟɦɨɣɲɢɯɬɵɞɨWɩɥ±ɬɟɦɩɟɪɚɬɭɪɵɩɥɚɜɥɟɧɢɹɤȾɠɤɝāɄ
Tɦ±ɫɤɪɵɬɚɹɬɟɩɥɨɬɚɩɥɚɜɥɟɧɢɹɦɟɬɚɥɥɚɤȾɠɤɝ
ɋɦ±ɫɪɟɞɧɹɹɬɟɩɥɨɟɦɤɨɫɬɶɦɟɬɚɥɥɚɜɢɧɬɟɪɜɚɥɟɬɟɦɩɟɪɚɬɭɪɨɬWɩɥɞɨɡɚɞɚɧɧɨɣ
ɬɟɦɩɟɪɚɬɭɪɵɦɟɬɚɥɥɚW
: =−++⋅−+=
ɦ
: =+⋅ −+⋅ −=
ɦ
Ɏɢɡɢɱɟɫɤɨɟɬɟɩɥɨɲɥɚɤɚɧɚɯɨɞɢɦɩɨɮɨɪɦɭɥɟ
()
ɤȾɠɤɝāɄ
ɜɵɩ
]
ȽȾɠ=
()
() ()
ȽȾɠ
=
: ȽȾɠ=+=
ɦ
:&W TP=⋅+⋅
ɲɥ ɲɥ ɲɥ ɲɥ ɲɥ
]
ɝɞɟ ɋɲɥ ± ɫɪɟɞɧɹɹ ɬɟɩɥɨɟɦɤɨɫɬɶ ɲɥɚɤɚ ɜ ɢɧɬɟɪɜɚɥɟ ɬɟɦɩɟɪɚɬɭɪɟ ɋ
ɤȾɠɤɝ⋅Ʉ
Tɲɥ±ɫɤɪɵɬɚɹɬɟɩɥɨɬɚɩɥɚɜɥɟɧɢɹɲɥɚɤɚɤȾɠɤɝPɲɥ±ɦɚɫɫɚɲɥɚɤɚɤɝ
()()
ɲɥ
: ȽȾɠ=+=
Ɍɨɝɞɚ
ɉɨɬɟɪɢɬɟɩɥɚɫɨɬɯɨɞɹɳɢɦɢɝɚɡɚɦɢ
′
: 9 L=⋅
ɝɞɟ9±ɨɛɴɟɦɜɵɞɟɥɹɸɳɢɯɫɹɝɚɡɨɜɦ
L±ɷɧɬɚɥɶɩɢɹɝɚɡɨɨɛɪɚɡɧɵɯɩɪɨɞɭɤɬɨɜɪɟɚɤɰɢɢɤȾɠɦ
ɇɚɯɨɞɢɦɷɧɬɚɥɶɩɢɸɩɪɨɞɭɤɬɨɜɪɟɚɤɰɢɢɞɥɹɤɚɠɞɨɝɨɢɡɝɚɡɨɜ
ɋɈ±ā
ɋɈ±ā
ɇ±ā
1±ā
L ɤȾɠɦ =
: ȽȾɠ=⋅=
Ɍɨɝɞɚ
ɉɨɬɟɪɢɫɨɯɥɚɠɞɚɸɳɟɣɜɨɞɨɣ
′
ɭɯ
:&*ȡǻWIJ=⋅⋅⋅⋅
′
ɪ
ɝɞɟɋɪ±ɭɞɟɥɶɧɚɹɢɡɨɛɚɪɧɚɹɬɟɩɥɨɟɦɤɨɫɬɶɜɨɞɵȾɠɤɝāɄ
*±ɪɚɫɯɨɞɜɨɞɵɦɱ
ȡ±ɩɥɨɬɧɨɫɬɶɜɨɞɵɩɪɢɬɟɦɩɟɪɚɬɭɪɟWɤɝɦ
WW
+
ɜɯ ɜɵɯ
ǻW
=
ɝɞɟǻW±ɪɚɡɧɨɫɬɶɬɟɦɩɟɪɚɬɭɪɧɚɜɵɯɨɞɟɢɧɚɜɯɨɞɟɋ
IJ±ɞɥɢɬɟɥɶɧɨɫɬɶɩɟɪɢɨɞɨɜɩɥɚɜɤɢɱ
ɂɫɯɨɞɧɵɟɞɚɧɧɵɟɩɪɟɞɫɬɚɜɥɟɧɵɜɬɚɛɥ
ɂɫɯɨɞɧɵɟɞɚɧɧɵɟ
Ɉɯɥɚɠɞɚɟɦɵɟ
ɷɥɟɦɟɧɬɵ
ɉɚɧɟɥɢɜɚɧɧɵ
ɉɚɧɟɥɢɫɜɨɞɚ
ɉɚɧɟɥɢɲɚɯɬɵ
ɍɞɟɪɠɢɜɚɸɳɢɟɩɚɥɶɰɵ
±ɩɟɪɢɨɞɧɚɝɪɟɜɚɢɩɥɚɜɥɟɧɢɹ−ɨɤɢɫɥɢɬɟɥɶɧɵɣɩɟɪɢɨɞ
Ɍɟɦɩɟɪɚɬɭɪɚɜɨɞɵɩɨɩɟɪɢɨɞɚɦɩɥɚɜɤɢ
ɧɚɜɯɨɞɟɋ ɧɚɜɵɯɨɞɟɋ
Ɍɨɝɞɚɩɨɬɟɪɢɫɨɯɥɚɠɞɚɸɳɟɣɜɨɞɨɣ
: Ⱦɠ
=⋅⋅ ⋅⋅= ⋅
ɜ
=+⋅++⋅=
Ɍ ɚ ɛ ɥ ɢ ɰ ɚ
ɋɪɟɞɧɢɣ
ɪɚɫɯɨɞ
ɜɨɞɵɦɱ
ȽȾɠ:
: Ⱦɠ
=⋅⋅ ⋅⋅= ⋅
ɫ
: Ⱦɠ
=⋅⋅ ⋅⋅= ⋅
ɲ
: Ⱦɠ
=⋅⋅ ⋅⋅= ⋅
ɩ
: ȽȾɠ=++ +⋅=
()
: Ⱦɠ
=⋅⋅ ⋅⋅= ⋅
ɜ
: Ⱦɠ
=⋅⋅ ⋅⋅= ⋅
ɫ
: Ⱦɠ
=⋅⋅ ⋅⋅= ⋅
ɲ
: Ⱦɠ
=⋅⋅ ⋅⋅= ⋅
ɩ
: ȽȾɠ=+++⋅=
()
: : : ȽȾɠ=+ = + =
Ɍɟɩɥɨɜɵɟɩɨɬɟɪɢɷɥɟɤɬɪɨɞɚɦɢɉɪɢɡɚɜɚɥɤɟɩɨɞɜɚɥɤɟɫɤɚɱɢɜɚɧɢɢɲɥɚɤɚ
ɢ ɩɨ ɞɪɭɝɢɦ ɩɪɢɱɢɧɚɦ ɷɥɟɤɬɪɨɞɵ ɧɚɯɨɞɹɬɫɹ ɜɧɟ ɪɚɛɨɱɟɝɨɩɪɨɫɬɪɚɧɫɬɜɚ ɩɟɱɢ
ɨɤɨɥɨɦɢɧɁɚɷɬɨɜɪɟɦɹɨɧɢɨɯɥɚɠɞɚɸɬɫɹɨɬɞɚɜɚɹɫɜɨɸɬɟɩɥɨɜɭɸɷɧɟɪɝɢɸ
ɨɤɪɭɠɚɸɳɟɣɫɪɟɞɟ
ɉɨɬɟɪɢɬɟɩɥɚɢɡɥɭɱɟɧɢɟɦɨɩɪɟɞɟɥɹɟɦɩɨɜɵɪɚɠɟɧɢɸ
ɉɨɬɟɪɢɬɟɩɥɚɤɨɧɜɟɤɰɢɟɣ
:1)ı7ɌIJ=⋅⋅⋅ − ⋅
ɢɡɥ ɷɥ ɜ
: Į ) 1 IJ W W =⋅⋅⋅⋅⋅−
ɤɨɧɜ ɷɥ ɜ
ɝɞɟα±ɤɨɷɮɮɢɰɢɟɧɬɬɟɩɥɨɨɬɞɚɱɢɤɤɚɥɦāɱāɋ
)±ɪɚɫɱɟɬɧɚɹɩɨɜɟɪɯɧɨɫɬɶɬɟɩɥɨɨɛɦɟɧɚɦ
()
()
1±ɤɨɥɢɱɟɫɬɜɨɷɥɟɤɬɪɨɞɨɜɲɬ
τ±ɜɪɟɦɹɧɚɯɨɠɞɟɧɢɹɷɥɟɤɬɪɨɞɨɜɜɧɟɩɟɱɢɱ
±ɫɪɟɞɧɹɹɬɟɦɩɟɪɚɬɭɪɚɷɥɟɤɬɪɨɞɚɋ
W
ɷɥ
±ɬɟɦɩɟɪɚɬɭɪɚɨɤɪɭɠɚɸɳɟɝɨɩɪɨɫɬɪɚɧɫɬɜɚɋ
W
ɜ
ȼɡɚɜɢɫɢɦɨɫɬɢɨɬɩɨɥɨɠɟɧɢɹɩɨɜɟɪɯɧɨɫɬɢɜɩɪɨɫɬɪɚɧɫɬɜɟɡɧɚɱɟɧɢɹαɦɟɧɹ
ɸɬɫɹ
ĮN W W
=⋅−
−
ɝɞɟ N ɞɥɹ ɝɨɪɢɡɨɧɬɚɥɶɧɨɣ ɩɨɜɟɪɯɧɨɫɬɢ ɨɛɪɚɳɟɧɧɨɣ ɜɧɢɡ ɧɚɩɪɢɦɟɪ
ɞɧɢɳɟɩɟɱɢ
ɞɥɹɜɟɪɬɢɤɚɥɶɧɨɣɩɨɜɟɪɯɧɨɫɬɢɤɨɠɭɯɫɬɟɧɷɥɟɤɬɪɨɞ
N
ɞɥɹɝɨɪɢɡɨɧɬɚɥɶɧɨɣɩɨɜɟɪɯɧɨɫɬɢɨɛɪɚɳɟɧɧɨɣɜɜɟɪɯɫɜɨɞɩɟɱɢ
N
Ɍɨɝɞɚɩɨɬɟɪɢɤɨɧɜɟɤɰɢɟɣ
ɋɥɟɞɨɜɚɬɟɥɶɧɨ
: Ⱦɠ
ɢɡɥ
: Ⱦɠ= ⋅⋅⋅⋅⋅⋅⋅−⋅ −= ⋅
ɤɨɧɜ
Ɍɟɩɥɨɭɧɨɫɢɦɨɟɱɚɫɬɢɰɚɦɢ)ɟɈ
ɉɨɬɟɪɢɢɡɥɭɱɟɧɢɟɦɩɪɢɨɬɤɪɵɬɨɦɫɜɨɞɟɩɟɱɢ
: ) 1 IJ W W W W =⋅⋅⋅⋅⋅−⋅−
ɤɨɧɜ ɷɥ ɜ ɷɥ ɜ
=⋅ ⋅ ⋅⋅ ⋅ ⋅ − ⋅ = ⋅
−
()
()
: : : ȽȾɠ=+ = + ⋅=
′
ɢɡɥ ɤɨɧɜ
: ȽȾɠ=⋅⋅+=
′
:ʌUıȥİIJ77=⋅ ⋅⋅⋅ ⋅⋅ −
:ʌUıȥİIJ77=⋅ ⋅⋅⋅ ⋅⋅ −
()
()
ɫɜ ɫɜ ɫɜ ɜ
ɫɜ ɜɚɧ ɜɚɧ ɜ
()
()
ɝɞɟUɫɜ±ɪɚɞɢɭɫɫɜɨɞɚɦ
σ±ɤɨɷɮɮɢɰɢɟɧɬɢɡɥɭɱɟɧɢɹɚɛɫɨɥɸɬɧɨɱɟɪɧɨɝɨɬɟɥɚȼɬɦ
āɄ
ψ±ɤɨɷɮɮɢɰɢɟɧɬɞɢɚɮɪɚɝɦɢɪɨɜɚɧɢɹ
±ɫɬɟɩɟɧɶɱɟɪɧɨɬɵɫɜɨɞɚɢɜɚɧɧɵ
ε
ɫɜεɜɚɧ
τ±ɩɪɨɞɨɥɠɢɬɟɥɶɧɨɫɬɶɪɚɛɨɬɵɫɨɬɤɪɵɬɵɦɫɜɨɞɨɦɫ
±ɫɪɟɞɧɹɹɬɟɦɩɟɪɚɬɭɪɚɜɧɭɬɪɟɧɧɟɣɩɨɜɟɪɯɧɨɫɬɢɫɜɨɞɚɩɟɱɢɋ
Ɍ
ɫɜ
Ɍɜ±ɬɟɦɩɟɪɚɬɭɪɚɜɧɟɲɧɟɣɩɨɜɟɪɯɧɨɫɬɢɋ
Ɍ
±ɫɪɟɞɧɹɹɬɟɦɩɟɪɚɬɭɪɚɩɨɜɟɪɯɧɨɫɬɢɜɚɧɧɵɩɪɢɜɵɩɭɫɤɟɋ
ɜɚɧ
ɝɞɟ=5ɇ±ɛɟɡɪɚɡɦɟɪɧɵɟɤɨɷɮɮɢɰɢɟɧɬɵ
= 5 +=+ + ⋅
ɉɨɞɫɬɚɜɢɦɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟɡɧɚɱɟɧɢɹɜɮɨɪɦɭɥɵ−ɩɨɥɭɱɢɦ
UU FRVș=⋅
șʌ
ș
=⋅ =
()
K
+
=
U
ɫɜ
U
5
=
K
ɫɜ
Į
=⋅
+5==
⋅⋅−−⋅−=ψ
U FRV =⋅ =
5
==
+
==
= =+ + ⋅ =
()
ȥ =− ⋅ − −⋅ ⋅ =
()
ɋɪɟɞɧɸɸɬɟɦɩɟɪɚɬɭɪɭɜɧɭɬɪɟɧɧɟɣɩɨɜɟɪɯɧɨɫɬɢɫɜɨɞɚɩɟɱɢɨɩɪɟɞɟɥɹɸɬɩɨ
ɮɨɪɦɭɥɟ
ɝɞɟU
±ɪɚɞɢɭɫɰɟɧɬɪɚɥɶɧɨɣɱɚɫɬɢɫɜɨɞɚɦ
ɰɫɜ
W
±ɫɪɟɞɧɹɹɬɟɦɩɟɪɚɬɭɪɚɫɜɨɞɨɜɵɯɩɚɧɟɥɟɣɋ
ɫɜɩɚɧ
W
±ɬɟɦɩɟɪɚɬɭɪɚɰɟɧɬɪɚɥɶɧɨɣɱɚɫɬɢɫɜɨɞɚɋ
ɰɫɜ
U U W W U
−⋅ +⋅
ɫɜ ɰɫɜ ɫɜɩɚɧ ɰɫɜ ɰɫɜ
W
=
ɫɜ
U
ɫɜ
ɂɫɯɨɞɧɵɟɞɚɧɧɵɟɞɥɹɪɚɫɱɟɬɚɩɪɟɞɫɬɚɜɥɟɧɵɜɬɚɛɥ
Ɍ ɚ ɛ ɥ ɢ ɰ ɚ
ɂɫɯɨɞɧɵɟɞɚɧɧɵɟ
Ɇɟɫɬɨɡɚɦɟɪɚɬɟɦɩɟɪɚɬɭɪɵ Ɍɟɦɩɟɪɚɬɭɪɚɋ
ɐɟɧɬɪɫɜɨɞɚɩɟɱɢ
ɉɚɧɟɥɢɫɜɨɞɚ
ɇɢɡɲɚɯɬɵ
Ⱦɧɨɜɚɧɧɵɩɨɫɥɟɜɵɩɭɫɤɚ
ɇɚɩɪɨɬɢɜɨɤɧɚɩɨɫɥɟɜɵɩɭɫɤɚ
Ɍɨɝɞɚɬɟɦɩɟɪɚɬɭɪɚɩɨɜɟɪɯɧɨɫɬɢɫɜɨɞɚɩɨɮɨɪɦɭɥɟ
W ɋ
ɋɪɟɞɧɹɹɬɟɦɩɟɪɚɬɭɪɚɩɨɜɟɪɯɧɨɫɬɢɜɚɧɧɵɩɪɢɜɵɩɭɫɤɟɦɟɬɚɥɥɚ
ɫɜ
ªº
«»
UWW
ɫɜ ɲɥ
«»
¬¼
=
W
ɜɚɧ
§· §·
UU
ɦɦ
−⋅+⋅
¨¸ ¨¸
©¹ ©¹
ɮɭɬɜ
U
ɫɜ
−⋅+⋅
==
ɝɞɟUɦ±ɪɚɞɢɭɫɦɟɬɚɥɥɢɱɟɫɤɨɣɜɚɧɧɵɦ
W
±ɬɟɦɩɟɪɚɬɭɪɚɮɭɬɟɪɨɜɤɢɋ
ɮɭɬɜ
Wɲɥ±ɬɟɦɩɟɪɚɬɭɪɚɲɥɚɤɚɋ
Ɍɟɦɩɟɪɚɬɭɪɚɩɨɜɟɪɯɧɨɫɬɢɜɚɧɧɵɩɨɮɨɪɦɭɥɟ
ªº
§· §·
«»
−⋅+⋅
¨¸ ¨¸
«»
©¹ ©¹
¬¼
==°
Wɋ
ɜɚɧ
Ɍɨɝɞɚɬɟɩɥɨɜɵɟɩɨɬɟɪɢɩɪɢ ɪɚɛɨɬɟɫ ɨɬɤɪɵɬɵɦɫɜɨɞɨɦɩɨɮɨɪɦɭɥɚɦ
ɫɨɫɬɚɜɹɬ
: Ⱦɠ
=⋅⋅⋅⋅ ⋅⋅⋅ − =⋅
: Ⱦɠ
=⋅⋅⋅⋅ ⋅⋅⋅ − = ⋅
−
−
: ȽȾɠ=+ ⋅=
′
()
()
()
Ɋɚɫɯɨɞɬɟɩɥɚɧɚɷɧɞɨɬɟɪɦɢɱɟɫɤɢɟɪɟɚɤɰɢɢɗɧɟɪɝɟɬɢɱɟɫɤɢɟɡɚɬɪɚɬɵɷɧɞɨ
ɬɟɪɦɢɱɟɫɤɢɯɩɪɨɰɟɫɫɨɜɫɜɹɡɚɧɵɫɧɚɝɪɟɜɨɦɢɢɫɩɚɪɟɧɢɟɦɜɥɚɝɢɜɧɟɫɟɧɧɨɣɡɚ
ɝɪɭɠɚɟɦɵɦɢɜɩɟɱɶɦɚɬɟɪɢɚɥɚɦɢɪɚɡɥɨɠɟɧɢɟɦɨɤɫɢɞɨɜɠɟɥɟɡɚɠɟɥɟɡɧɨɣɪɭɞɵ
ɤɚɪɛɨɧɚɬɨɜɤɚɥɶɰɢɹɩɪɢ ɧɟɞɨɩɚɥɟɢɡɜɟɫɬɢɢɥɢ ɝɢɞɪɚɬɨɜɨɤɫɢɞɚɤɚɥɶɰɢɹ ɩɪɢ
ɝɢɞɪɚɬɚɰɢɢɢɡɜɟɫɬɢɲɥɚɤɨɨɛɪɚɡɨɜɚɧɢɟɦɂɡɪɟɚɤɰɢɣɩɪɨɢɫɯɨɞɹɳɢɯɜɩɟɱɢɷɧ
ɞɨɬɟɪɦɢɱɟɫɤɢɦɢɹɜɥɹɸɬɫɹɪɟɚɤɰɢɢɞɟɫɭɥɶɮɭɪɚɰɢɢɨɛɟɡɭɝɥɟɪɨɠɢɜɚɧɢɹɜɨɫɫɬɚ
ɧɨɜɥɟɧɢɹɢɩɪɨɰɟɫɫɵɢɫɩɚɪɟɧɢɹɷɥɟɦɟɧɬɨɜȾɥɹɛɨɥɟɟɨɛɳɟɣɨɰɟɧɤɢɷɧɞɨɬɟɪɦɢ
ɱɟɫɤɢɯɪɟɚɤɰɢɣɩɪɢɧɢɦɚɟɦ ɭɞɟɥɶɧɵɟɷɧɟɪɝɟɬɢɱɟɫɤɢɟɡɚɬɪɚɬɵ ɧɚɷɧɞɨɬɟɪɦɢɱɟ
ɫɤɢɟɩɪɨɰɟɫɫɵɪɚɜɧɵɦɢɆȾɠɬɬɨɝɞɚ
ɝɞɟPR±ɟɦɤɨɫɬɶɩɟɱɢɬ
:P=⋅
: ȽȾɠ=⋅=
′
!
′
()
Ɉ
Ɍɟɩɥɨɜɵɟɩɨɬɟɪɢɷɥɟɤɬɪɢɱɟɫɤɨɣɫɟɬɶɸɗɬɢɩɨɬɟɪɢɫɜɹɡɚɧɵɫɬɟɦɱɬɨɩɪɢ
ɩɪɨɯɨɠɞɟɧɢɢɬɨɤɚɩɨɩɪɨɜɨɞɧɢɤɭɜɧɟɦɜɵɞɟɥɹɟɬɫɹɬɟɩɥɨɤɨɬɨɪɨɟɱɚɫɬɢɱɧɨɪɚ
ɫɫɟɢɜɚɟɬɫɹɜɨɤɪɭɠɚɸɳɟɟɩɪɨɫɬɪɚɧɫɬɜɨɗɥɟɤɬɪɢɱɟɫɤɢɟɩɨɬɟɪɢɡɚɜɢɫɹɬɨɬɤɨɧɫ
ɬɪɭɤɰɢɢɬɨɤɨɩɨɞɜɨɞɚɩɚɪɚɦɟɬɪɨɜɷɥɟɤɬɪɢɱɟɫɤɨɝɨɪɟɠɢɦɚɢɫɬɚɛɢɥɶɧɨɫɬɢɝɨɪɟ
ɧɢɹɷɥɟɤɬɪɢɱɟɫɤɢɯɞɭɝȼ ɫɨɫɬɚɜɷɥɟɤɬɪɢɱɟɫɤɢɯɩɨɬɟɪɶɜɯɨɞɹɬɩɨɬɟɪɢ ɜɲɢɧɚɯ
ɜɵɫɨɤɨɝɨɧɚɩɪɹɠɟɧɢɹɜɬɪɚɧɫɮɨɪɦɚɬɨɪɟɜɨɜɫɩɨɦɨɝɚɬɟɥɶɧɨɣɚɩɩɚɪɚɬɭɪɟɜɤɨ
ɪɨɬɤɨɣɫɟɬɢ
ɉɨɡɚɤɨɧɭȾɠɨɭɥɹ±Ʌɟɧɰɚ
:,5IJ=⋅⋅′
ɝɞɟ,±ɫɢɥɚɬɨɤɚȺɩɨɞɚɧɧɵɦɩɟɱɢ, ɤȺ
5±ɫɨɩɪɨɬɢɜɥɟɧɢɟɈɦ
IJ±ɜɪɟɦɹɪɚɛɨɬɵɩɟɱɢɩɨɞɬɨɤɨɦɫ
Ɍɨɝɞɚ
: ȽȾɠ
=⋅⋅⋅=′
−
Ɉɞɧɚɤɨɧɟɜɫɟɬɟɩɥɨɪɚɫɫɟɢɜɚɟɬɫɹɜɩɪɨɫɬɪɚɧɫɬɜɟɢɩɨɷɬɨɦɭɬɨɱɧɨɟɨɩɪɟɞɟ
ɥɟɧɢɟɷɬɨɣɞɨɥɢɬɟɩɥɨɜɵɯɩɨɬɟɪɶɡɚɬɪɭɞɧɟɧɨ
ɉɨɬɟɪɢɬɟɩɥɚɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɶɸɱɟɪɟɡɮɭɬɟɪɨɜɤɭɈɩɪɟɞɟɥɢɦɩɨɬɟɪɢɬɟ
ɩɥɚɱɟɪɟɡɩɨɞɢɧɭɩɟɱɢɎɭɬɟɪɨɜɤɚɩɨɞɢɧɵɫɨɫɬɨɢɬɢɡ ɦɚɝɧɟɡɢɬɨɜɨɣɧɚɛɢɜɤɢ±
ɦɦ ɢ ɦɚɝɧɟɡɢɬɨɜɨɣ ɮɭɬɟɪɨɜɤɢ − ɦɦ ɩɪɢɧɢɦɚɟɦ ɢɯ ɤɚɤ ɨɞɢɧ ɫɥɨɣ
ɬɨɥɳɢɧɨɣɦɦɢɲɚɦɨɬɚɥɟɝɤɨɜɟɫɚɬɨɥɳɢɧɨɣɦɦɌɟɦɩɟɪɚɬɭɪɚɜɧɭɬɪɟɧ
ɧɟɣɩɨɜɟɪɯɧɨɫɬɢɩɨɞɢɧɵ W ɋ ɬɟɦɩɟɪɚɬɭɪɚɧɚɪɭɠɧɨɣ ɩɨɜɟɪɯɧɨɫɬɢɤɨ
ɠɭɯɚW ɋɄɨɷɮɮɢɰɢɟɧɬɵɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ
()
ɄɦȼɬW
ɆɆ
ɒɒ
⋅⋅−=λ
()
ɄɦȼɬW
⋅⋅+=λ
ɉɪɢɧɢɦɚɹɜɩɟɪɜɨɦɩɪɢɛɥɢɠɟɧɢɢɥɢɧɟɣɧɨɟɪɚɫɩɪɟɞɟɥɟɧɢɟɬɟɦɩɟɪɚɬɭɪɵɩɨ
ɬɨɥɳɢɧɟɮɭɬɟɪɨɜɤɢɧɚɯɨɞɢɦɬɟɦɩɟɪɚɬɭɪɭɧɚɝɪɚɧɢɰɟɪɚɡɞɟɥɚɫɥɨɟɜ
WWWW
=+ + ⋅
−
ɦɲ
į
ɲ
įį
+
ɲɦ
ɝɞɟįɦįɲ±ɬɨɥɳɢɧɚɦɚɝɧɟɡɢɬɨɜɨɣɮɭɬɟɪɨɜɤɢɢɲɚɦɨɬɚɦ
W W ± ɬɟɦɩɟɪɚɬɭɪɚ ɜɧɭɬɪɟɧɧɟɣ ɢ ɧɚɪɭɠɧɨɣ ɩɨɜɟɪɯɧɨɫɬɢ ɮɭɬɟɪɨɜɤɢ ɩɨ
ɞɢɧɵɋ
W ɋ
=+ − ⋅ =°
ɦɲ
−
Ɍɨɝɞɚɤɨɷɮɮɢɰɢɟɧɬɵɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢɦɚɬɟɪɢɚɥɨɜ
Ɇ
ɒ
()
⋅−=λ
⋅−=λ
+
+
+
()
Ʉɦȼɬ
⋅=
()
Ʉɦȼɬ
⋅=
ɉɨɮɨɪɦɭɥɟɤɨɧɜɟɤɬɢɜɧɨɝɨɬɟɩɥɨɨɛɦɟɧɚɩɪɢɦɟɧɹɟɦɨɣɞɥɹɩɪɢɛɥɢɠɟɧɧɵɯ
ɪɚɫɱɟɬɨɜ
ĮW=+ ⋅
ɫɬ
ɝɞɟWɫɬ±ɬɟɦɩɟɪɚɬɭɪɚɤɨɠɭɯɚɪɚɜɧɚW&
ɉɪɢW ɋɤɨɷɮɮɢɰɢɟɧɬɬɟɩɥɨɨɬɞɚɱɢɤɨɧɜɟɤɰɢɟɣɨɛɪɚɳɟɧɧɨɣɜɧɢɡɩɨ
ɜɟɪɯɧɨɫɬɢɪɚɜɟɧ
ɋɨɝɥɚɫɧɨɮɨɪɦɭɥɟɩɥɨɬɧɨɫɬɶɬɟɩɥɨɜɨɝɨɩɨɬɨɤɚɱɟɪɟɡɞɜɭɯɫɥɨɣɧɭɸɫɬɟɧɤɭ
T ȼɬɦ
ɩɨɞ
WW WW
== = =
į
L
+
¦
ȜĮ
L
Į Į W =⋅=⋅+ ⋅
ɤɨɧɜ
Į ȼɬ ɦ Ʉ =⋅+ ⋅ = ⋅
ɤɨɧɜ
ɰɰ
ɤɨɧɜL
()
įį
ɦɲ
++
ȜȜĮ
ɦɲɤɨɧɜ
()
++
()
−
ɝɞɟWɰ±ɬɟɦɩɟɪɚɬɭɪɚɰɟɯɚɋ
įL±ɬɨɥɳɢɧɚɭɱɚɫɬɤɨɜɫɬɟɧɦ
ɍɬɨɱɧɢɦɡɧɚɱɟɧɢɟɬɟɦɩɟɪɚɬɭɪɵɧɚɝɪɚɧɢɰɟɪɚɡɞɟɥɚɫɥɨɟɜɮɭɬɟɪɨɜɤɢ
ɪ
WWT
=− ⋅
−
ɦɲ ɩɨɞ
WW
=+
ɰ
į
ɦ
Ȝ
ɦ
T
ɩɨɞ
Į
ɤɨɧɜ
+
+
ɪ
W ɋ
=− ⋅ = °
ɦɲ
−
W ɋ
Ȝ ȼɬɦɄ
Ɍɨɝɞɚ
ɇɚɣɞɟɦɭɬɨɱɧɟɧɧɨɟɡɧɚɱɟɧɢɟɩɥɨɬɧɨɫɬɢɩɨɬɨɤɚɬɟɩɥɚɱɟɪɟɡɩɨɞɢɧɭ
=− ⋅ = ⋅
ɦ
Ȝ ȼɬ ɦ Ʉ
=+ ⋅ = ⋅
ɲ
Į ȼɬ ɦ Ʉ=⋅+ ⋅ = ⋅
ɤɨɧɜ
T ȼɬɦ
ɩɨɞ
()
==
=+ = °
()
()
−
++
Ɍɟɩɟɪɶɦɨɠɧɨɧɚɣɬɢɭɬɨɱɧɟɧɧɵɟɡɧɚɱɟɧɢɹɬɟɦɩɟɪɚɬɭɪɧɚɝɪɚɧɢɰɟɪɚɡɞɟɥɚ
ɫɥɨɟɜɫɪɟɞɧɢɟɬɟɦɩɟɪɚɬɭɪɵɫɥɨɟɜɢɤɨɷɮɮɢɰɢɟɧɬɵɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ
Ɍɨɝɞɚ
W ɋ
=− ⋅ = °
ɦɲ
−
W ɋ
Ȝ ȼɬ ɦ Ʉ
=− ⋅ = ⋅
ɦ
Ȝ ȼɬ ɦ Ʉ
=+ ⋅ = ⋅
ɲ
Į ȼɬ ɦ Ʉ=⋅+ ⋅ = ⋅
ɤɨɧɜ
T ȼɬɦ
==
ɩɨɞ
=+ = °
()
()
()
−
++
+
+
ɉɨɫɤɨɥɶɤɭɪɚɫɯɨɠɞɟɧɢɟ ɦɟɠɞɭ ɞɜɭɦɹɩɨɫɥɟɞɧɢɦɢ ɡɧɚɱɟɧɢɹɦɢɩɥɨɬɧɨɫɬɢ
ɬɟɩɥɨɜɨɝɨɩɨɬɨɤɚɱɟɪɟɡɩɨɞɫɨɫɬɚɜɥɹɟɬɦɟɧɟɟ
−−
T T
⋅= ⋅= <
T
ɬɨɩɨɫɥɟɞɧɟɟɡɧɚɱɟɧɢɟ T ȼɬɦɫɱɢɬɚɟɦɨɤɨɧɱɚɬɟɥɶɧɵɦ>ɫ@
Ɍɨɝɞɚɩɨɬɟɪɢɬɟɩɥɚɱɟɪɟɡɩɨɞɢɧɭɧɚɯɨɞɢɦɩɨɮɨɪɦɭɥɟ
:T)IJ=⋅⋅
′
ɩɨɞ
ɝɞɟ)±ɩɥɨɳɚɞɶɧɚɪɭɠɧɨɣɩɨɜɟɪɯɧɨɫɬɢɩɨɞɢɧɵɦ
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