Лекции по классической механике. Учебное пособие
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Лекция10.ОСНОВЫМЕХАНИКИГАМИЛЬТОНА
ɉɨɥɧɚɹ ɩɪɨɢɡɜɨɞɧɚɹ ɩɨ ɜɪɟɦɟɧɢ ɨɬ ɜɟɥɢɱɢɧɵ f (qk pk t) ɨɩɪɟɞɟɥɹɟɬɫɹ ɮɨɪɦɭɥɨɣ ɂɡ ɧɟɟ ɫɥɟɞɭɟɬ ɱɬɨɜɟɥɢɱɢɧɚ f ɫɨɯɪɚɧɹɟɬɫɹ ɹɜɥɹɟɬɫɹ ɢɧɬɟ ɝɪɚɥɨɦ ɞɜɢɠɟɧɢɹ ɟɫɥɢ ɨɧɚ ɹɜɧɨ ɧɟ ɡɚɜɢɫɢɬ ɨɬ ɜɪɟɦɟɧɢ ɢ ɟɟ ɫɤɨɛɤɚ ɉɭɚɫɫɨɧɚ ɫ ɮɭɧɤɰɢɟɣ Ƚɚɦɢɥɶɬɨɧɚ ɨɛɪɚɳɚɟɬɫɹ ɜ ɧɭɥɶ
ɉɭɫɬɶ ɜɟɥɢɱɢɧɵ f g ɹɜɥɹɸɬɫɹ ɫɨɯɪɚɧɹɸɳɢɦɢɫɹ ɜɟɥɢɱɢɧɚɦɢ ɋɨɫɬɚɜɢɦ ɢɡ ɧɢɯ ɫɤɨɛɤɭ ɉɭɚɫɫɨɧɚ{f g}Ⱦɚɧɧɚɹ ɜɟɥɢɱɢɧɚ ɹɜɥɹɟɬɫɹ ɫɨɯɪɚɧɹɸɳɟɣɫɹ ɜɟɥɢ ɱɢɧɨɣ ɬ ɟ ɹɜɥɹɟɬɫɹ ɢɧɬɟɝɪɚɥɨɦ ɞɜɢɠɟɧɢɹ {f g}= const ɬɟɨɪɟɦɚ ɉɭɚɫɫɨɧɚ ȿɫɥɢ ɞɚɧɧɵɟ ɫɤɨɛɤɢ ɉɭɚɫɫɨɧɚ ɧɟ ɫɜɨɞɹɬɫɹ ɤ ɧɭɥɸ ɢɥɢ ɤ ɤɚɤɨɣ ɥɢɛɨ ɤɨɦɛɢɧɚ ɰɢɢ ɮɭɧɤɰɢɣ f g ɬɨ ɨɧɢ ɞɚɸɬ ɧɨɜɵɣ ɢɧɬɟɝɪɚɥ ɞɜɢɠɟɧɢɹ
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ɅȿɄɐɂə ȽȺɊɆɈɇɂɑȿɋɄɂȿ ɄɈɅȿȻȺɇɂə ɆȿɏȺɇɂɑȿɋɄɂɏ
ɋɂɋɌȿɆ ɋ ɈȾɇɈɃ ɋɌȿɉȿɇɖɘ ɋȼɈȻɈȾɕ
Ʌɢɧɟɣɧɵɣ ɝɚɪɦɨɧɢɱɟɫɤɢɣ ɨɫɰɢɥɥɹɬɨɪ Ɏɭɧɤɰɢɹ Ʌɚɝɪɚɧɠɚ
Ɋɚɫɫɦɨɬɪɢɦ ɦɟɯɚɧɢɱɟɫɤɭɸ ɫɢɫɬɟɦɭ ɫ ɨɞɧɨɣ ɫɬɟɩɟɧɶɸ ɫɜɨɛɨɞɵ ɫɨ ɫɬɚɰɢɨ
ɧɚɪɧɵɦɢ ɩɨɬɟɧɰɢɚɥɶɧɵɦɢ ɫɢɥɚɦɢ ɧɚ ɤɨɬɨɪɭɸ ɧɚɤɥɚɞɵɜɚɸɬɫɹ ɢɞɟɚɥɶɧɵɟ ɫɬɚ
ɰɢɨɧɚɪɧɵɟ ɫɜɹɡɢ ɋɱɢɬɚɟɦ ɱɬɨ ɫɢɫɬɟɦɚ ɢɦɟɟɬ ɫɨɫɬɨɹɧɢɟ ɭɫɬɨɣɱɢɜɨɝɨ ɪɚɜɧɨɜɟ ɫɢɹ q ɜ ɤɨɬɨɪɨɦ ɩɨɬɟɧɰɢɚɥɶɧɚɹ ɷɧɟɪɝɢɹ ɫɢɫɬɟɦɵ ɩɪɢɧɢɦɚɟɬ ɦɢɧɢɦɚɥɶɧɨɟ
ɡɧɚɱɟɧɢɟ ɉɭɫɬɶx = q − q ± ɫɦɟɳɟɧɢɟ ɫɢɫɬɟɦɵ ɢɡ ɫɨɫɬɨɹɧɢɹ ɭɫɬɨɣɱɢɜɨɝɨ ɪɚɜɧɨ ɜɟɫɢɹ Ɋɚɡɥɨɠɢɦ ɮɭɧɤɰɢɸ ɩɨɬɟɧɰɢɚɥɶɧɨɣ ɷɧɟɪɝɢɢU (q)ɜ ɪɹɞ ɩɨ ɫɦɟɳɟɧɢɹɦx
U(q)= U(q )+ ¨dU |
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ȼ ɞɚɧɧɨɦ ɜɵɪɚɠɟɧɢɢ ɞɥɹ ɩɨɬɟɧɰɢɚɥɶɧɨɣ ɷɧɟɪɝɢɢ ɫɢɫɬɟɦɵ ɦɵ ɨɝɪɚɧɢɱɢ
ɜɚɟɦɫɹ ɬɨɱɧɨɫɬɶɸ ɞɨ ɜɟɥɢɱɢɧ ɜɬɨɪɨɝɨ ɩɨɪɹɞɤɚ ɦɚɥɨɫɬɢ Ⱦɚɧɧɨɟ ɩɪɢɛɥɢɠɟɧɢɟ ɧɨɫɢɬ ɧɚɡɜɚɧɢɟɝɚɪɦɨɧɢɱɟɫɤɨɝɨ ɩɪɢɛɥɢɠɟɧɢɹɉɪɨɜɟɞɟɦ ɧɨɪɦɢɪɨɜɤɭ ɩɨɬɟɧɰɢ ɚɥɶɧɨɣ ɷɧɟɪɝɢɢ ɩɨɥɨɠɢɜ ɡɧɚɱɟɧɢɟU (q )ɪɚɜɧɵɦ ɧɭɥɸ ɍɫɥɨɜɢɟ ɦɢɧɢɦɭɦɚ ɩɨ
ɬɟɧɰɢɚɥɶɧɨɣ ɷɧɟɪɝɢɢ ɬɪɟɛɭɟɬ
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ɉɨɫɬɨɹɧɧɚɹ ɜɟɥɢɱɢɧɚ k ɧɚɡɵɜɚɟɬɫɹ ɫɢɥɨɜɵɦ ɤɨɷɮɮɢɰɢɟɧɬɨɦ ɫɢɫɬɟɦɵ ɋ ɭɱɟɬɨɦ ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɩɨɬɟɧɰɢɚɥɶɧɨɣ ɷɧɟɪɝɢɢ ɤɨɥɟɛɚɬɟɥɶɧɨɣ ɫɢɫɬɟɦɵ ɜ ɝɚɪɦɨɧɢɱɟɫɤɨɦ ɩɪɢɛɥɢɠɟɧɢɢ ɢɦɟɟɬ ɜɢɞ
U(x)= k x
Ʉɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɫɢɫɬɟɦɵ ɢɦɟɟɬ ɜɢɞ
= m
T x
ɝɞɟ ɜɟɥɢɱɢɧɚm ɟɫɬɶ ɤɢɧɟɦɚɬɢɱɟɫɤɢɣ ɩɚɪɚɦɟɬɪ ɫɢɫɬɟɦɵ ɜ ɱɚɫɬɧɨɦ ɫɥɭɱɚɟ ɦɚɫ ɫɚ ɤɨɥɟɛɥɸɳɟɣɫɹ ɱɚɫɬɢɰɵ
Ɏɭɧɤɰɢɹ Ʌɚɝɪɚɧɠɚ ɫɢɫɬɟɦɵ ɫɨɜɟɪɲɚɸɳɟɣ ɝɚɪɦɨɧɢɱɟɫɤɢɟ ɤɨɥɟɛɚɧɢɹ ɩɪɟɞɫɬɚɜɥɹɟɬɫɹ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ
L= m x − k x
ɋɢɫɬɟɦɚ ɮɭɧɤɰɢɹ Ʌɚɝɪɚɧɠɚ ɤɨɬɨɪɨɣ ɢɦɟɟɬ ɜɢɞ ɧɚɡɵɜɚɟɬɫɹɥɢɧɟɣ ɧɵɦ ɝɚɪɦɨɧɢɱɟɫɤɢɦ ɨɫɰɢɥɥɹɬɨɪɨɦ
ɋɨɫɬɚɜɢɜ ɭɪɚɜɧɟɧɢɟ Ʌɚɝɪɚɧɠɚ ɞɥɹ ɥɢɧɟɣɧɨɝɨ ɝɚɪɦɨɧɢɱɟɫɤɨɝɨ ɨɫɰɢɥɥɹɬɨ
ɪɚ ɩɨɥɭɱɢɦ ɭɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɫɢɫɬɟɦɵ ɜɛɥɢɡɢ ɩɨɥɨɠɟɧɢɹ ɭɫɬɨɣɱɢɜɨɝɨ ɫɨ
ɫɬɨɹɧɢɹ
m x + k x =
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Лекция11.ГАРМОНИЧЕСКИЕКОЛЕБАНИЯМЕХАНИЧЕСКИХСИСТЕМСОДНОЙСТЕПЕНЬЮСВОБОДЫ
Ɋɚɡɞɟɥɢɦ ɩɪɚɜɭɸ ɢ ɥɟɜɭɸ ɱɚɫɬɶ ɧɚm ɢ ɜɜɟɞɟɦ ɨɛɨɡɧɚɱɟɧɢɟ
mk = ω
ȼɟɥɢɱɢɧɚω ɧɚɡɵɜɚɟɬɫɹɫɨɛɫɬɜɟɧɧɨɣ ɱɚɫɬɨɬɨɣ ɫɢɫɬɟɦɵɫɨɜɟɪɲɚɸɳɟɣ ɝɚɪ ɦɨɧɢɱɟɫɤɢɟ ɤɨɥɟɛɚɧɢɹ ɩɨɞ ɞɟɣɫɬɜɢɟɦ ɨɛɨɛɳɟɧɧɨɣ ɤɜɚɡɢɭɩɪɭɝɨɣ ɫɢɥɵQ = −kx
ɍɪɚɜɧɟɧɢɟ ɹɜɥɹɟɬɫɹ ɨɞɧɨɪɨɞɧɵɦ ɥɢɧɟɣɧɵɦ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɦ ɭɪɚɜɧɟɧɢɟɦɜɬɨɪɨɝɨɩɨɪɹɞɤɚɫ ɩɨɫɬɨɹɧɧɵɦɢɤɨɷɮɮɢɰɢɟɧɬɚɦɢȿɝɨɨɛɳɟɟɪɟ ɲɟɧɢɟ ɢɦɟɟɬ ɜɢɞ
x(t)= AFRV(ω t + α )
Ɂɞɟɫɶ A ± ɦɚɤɫɢɦɚɥɶɧɨɟ ɫɦɟɳɟɧɢɟ ɢɡ ɩɨɥɨɠɟɧɢɹ ɪɚɜɧɨɜɟɫɢɹ ± ɚɦɩɥɢɬɭɞɚ ɤɨɥɟɛɚɧɢɣ(ω t + α )± ɮɚɡɚ ɤɨɥɟɛɚɧɢɣα ± ɧɚɱɚɥɶɧɚɹ ɮɚɡɚ ɤɨɥɟɛɚɧɢɣ Ⱥɦɩɥɢ
ɬɭɞɚ ɤɨɥɟɛɚɧɢɣ ɢ ɧɚɱɚɥɶɧɚɹ ɮɚɡɚ ɨɩɪɟɞɟɥɹɸɬɫɹ ɢɡ ɧɚɱɚɥɶɧɵɯ ɭɫɥɨɜɢɣ ɋɨɛɫɬ
ɜɟɧɧɚɹ ɱɚɫɬɨɬɚ ɤɨɥɟɛɚɧɢɣ ɧɟ ɡɚɜɢɫɢɬ ɨɬ ɧɚɱɚɥɶɧɵɯ ɭɫɥɨɜɢɣ ɢ ɨɩɪɟɞɟɥɹɟɬɫɹ ɩɚ
ɪɚɦɟɬɪɚɦɢ ɫɢɫɬɟɦɵ
Ɏɭɧɤɰɢɹ Ƚɚɦɢɥɶɬɨɧɚ Ɏɚɡɨɜɵɟ ɬɪɚɟɤɬɨɪɢɢ
Ɉɛɨɛɳɟɧɧɵɣ ɢɦɩɭɥɶɫ ɥɢɧɟɣɧɨɝɨ ɝɚɪɦɨɧɢɱɟɫɤɨɝɨ ɨɫɰɢɥɥɹɬɨɪɚ
p = ∂∂Lx = mx
Ɏɭɧɤɰɢɹ Ƚɚɦɢɥɶɬɨɧɚ
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Ⱦɚɧɧɚɹ ɮɭɧɤɰɢɹ ɜɵɪɚɠɟɧɚ ɱɟɪɟɡ ɨɛɨɛɳɟɧɧɵɣ ɢɦɩɭɥɶɫ ɢ ɨɛɨɛɳɟɧɧɭɸ ɤɨɨɪɞɢɧɚɬɭ ± ɫɦɟɳɟɧɢɟ ɱɚɫɬɢɰɵ ɨɬ ɩɨɥɨɠɟɧɢɹ ɭɫɬɨɣɱɢɜɨɝɨ ɪɚɜɧɨɜɟɫɢɹ Ɏɭɧɤ ɰɢɹ Ƚɚɦɢɥɶɬɨɧɚ ɹɜɧɨ ɧɟ ɡɚɜɢɫɢɬ ɨɬ ɜɪɟɦɟɧɢ ɢ ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɩɨɥɧɭɸ ɦɟ ɯɚɧɢɱɟɫɤɭɸ ɷɧɟɪɝɢɸ ɫɢɫɬɟɦɵ
E = p + k x
m
ɤɨɬɨɪɚɹ ɫɨɯɪɚɧɹɟɬɫɹ ɜ ɩɪɨɰɟɫɫɟ ɞɜɢɠɟɧɢɹ ɉɟɪɟɩɢɲɟɦ ɮɨɪɦɭɥɭ ɜ ɜɢɞɟ
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Ʉɨɨɪɞɢɧɚɬɚɦɢ ɮɚɡɨɜɨɝɨ ɩɪɨɫɬɪɚɧɫɬɜɚ ɹɜɥɹɸɬɫɹ ɩɟɪɟɦɟɧɧɵɟ ɫɨɫɬɨɹɧɢɹ ɫɢɫɬɟɦɵx ɢ p ɍɪɚɜɧɟɧɢɟ ɜ ɮɚɡɨɜɨɦ ɩɪɨɫɬɪɚɧɫɬɜɟ ɹɜɥɹɟɬɫɹ ɭɪɚɜɧɟɧɢɟɦ ɷɥɥɢɩɫɚ Ɍɚɤɢɦ ɨɛɪɚɡɨɦ ɮɚɡɨɜɨɣ ɬɪɚɟɤɬɨɪɢɟɣ ɫɢɫɬɟɦɵ ɹɜɥɹɟɬɫɹ ɷɥɥɢɩɫ
ȼɵɩɢɲɟɦ ɤɚɧɨɧɢɱɟɫɤɢɟ ɭɪɚɜɧɟɧɢɹ Ƚɚɦɢɥɶɬɨɧɚ ɥɢɧɟɣɧɨɝɨ ɝɚɪɦɨɧɢɱɟɫɤɨ
ɝɨ ɨɫɰɢɥɥɹɬɨɪɚ
dpdt = − ∂∂Hx = − kx dxdt = ∂∂Hp = x
ɉɟɪɜɨɟɭɪɚɜɧɟɧɢɟɟɫɬɶɡɚɤɨɧɢɡɦɟɧɟɧɢɹɢɦɩɭɥɶɫɚɱɚɫɬɢɰɵȿɝɨɦɨɠɧɨ ɩɟɪɟɩɢɫɚɬɶ ɫ ɭɱɟɬɨɦ ɨɩɪɟɞɟɥɟɧɢɹ ɨɛɨɛɳɟɧɧɨɝɨ ɢɦɩɭɥɶɫɚ ɜ ɜɢɞɟ
63
М. А. Михайлов. ЛЕКЦИИ ПО КЛАССИЧЕСКОЙ МЕХАНИКЕ
mx = − kx
ɬ ɟɜ ɜɢɞɟɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨɭɪɚɜɧɟɧɢɹɜɬɨɪɨɝɨɩɨɪɹɞɤɚ±ɜɬɨɪɨɝɨɡɚɤɨɧɚ ɇɶɸɬɨɧɚ ȼɬɨɪɨɟ ɭɪɚɜɧɟɧɢɟ ɹɜɥɹɟɬɫɹ ɨɩɪɟɞɟɥɟɧɢɟɦ ɫɤɨɪɨɫɬɢ
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ ɤɚɧɨɧɢɱɟɫɤɢɟ ɭɪɚɜɧɟɧɢɹȽɚɦɢɥɶɬɨɧɚɩɨɥɧɨɫɬɶɸ ɨɩɢɫɵɜɚɸɬ ɷɜɨɥɸɰɢɸ ɫɢɫɬɟɦɵ ɢ ɩɪɢ ɡɚɞɚɧɧɵɯ ɧɚɱɚɥɶɧɵɯ ɭɫɥɨɜɢɹɯ ɢɦɟɸɬ ɟɞɢɧɫɬɜɟɧɧɨɟ ɪɟɲɟɧɢɟ
Ɂɚɞɚɧɢɹɤɨɨɪɞɢɧɚɬɵɢ ɢɦɩɭɥɶɫɚɱɚɫɬɢɰɵɩɨɥɧɨɫɬɶɸɨɩɪɟɞɟɥɹɸɬɮɭɧɤ ɰɢɸ Ƚɚɦɢɥɶɬɨɧɚ ɫɢɫɬɟɦɵ ɤɨɬɨɪɚɹ ɤɚɤ ɢ ɮɭɧɤɰɢɹ Ʌɚɝɪɚɧɠɚ ɹɜɥɹɟɬɫɹ ɮɭɧɤɰɢɟɣ ɫɨɫɬɨɹɧɢɹ ɫɢɫɬɟɦɵ
ɉɪɢɦɟɪ Ɇɚɬɟɦɚɬɢɱɟɫɤɢɣ ɦɚɹɬɧɢɤ ɞɥɢɧɵ l ɫ ɦɚɫɫɨɣ m Ɂɚɩɢɲɟɦ ɮɭɧɤɰɢɸ Ʌɚɝɪɚɧɠɚ ɫɢɫɬɟɦɵ ɜɵɛɪɚɜ ɜ ɤɚɱɟɫɬɜɟ ɨɛɨɛɳɟɧɧɨɣ ɤɨɨɪɞɢɧɚɬɵ ɭɝɨɥ ɨɬɤɥɨɧɟɧɢɹϕ ɨɬ ɩɨɥɨɠɟɧɢɹ ɪɚɜɧɨɜɟɫɢɹ ɪɢɫ Ɏɭɧɤɰɢɹ Ʌɚɝɪɚɧɠɚ
L= ml ϕ + m g l FRVϕ ɋɱɢɬɚɹ ɱɬɨ ɨɬɤɥɨɧɟɧɢɹ ɫɢɫɬɟɦɵ ɨɬ ɩɨɥɨɠɟɧɢɹ ɪɚɜɧɨɜɟ
ɫɢɹ ɦɚɥɵ ɡɚɩɢɲɟɦ ɮɭɧɤɰɢɸ Ʌɚɝɪɚɧɠɚ ɜ ɝɚɪɦɨɧɢɱɟɫɤɨɦ ɩɪɢɛɥɢɠɟɧɢɢ
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L = ml ϕ − mgl ϕ
ɦɵɨɩɭɫɬɢɥɢɩɨɫɬɨɹɧɧɭɸɜɟɥɢɱɢɧɭɤɨɬɨɪɚɹɫɜɹɡɚɧɚɫ ɩɟɪɟɧɨɪɦɢɪɨɜɤɨɣɩɨ ɬɟɧɰɢɚɥɶɧɨɣ ɷɧɟɪɝɢɢ
ɋɨɫɬɚɜɢɦ ɭɪɚɜɧɟɧɢɟ Ʌɚɝɪɚɧɠɚ ml ϕ + mglϕ = ɢɥɢ ϕ + gl ϕ = Ⱦɚɧɧɨɟ
ɭɪɚɜɧɟɧɢɟ ɟɫɬɶ ɭɪɚɜɧɟɧɢɟ ɝɚɪɦɨɧɢɱɟɫɤɢɯ ɤɨɥɟɛɚɧɢɣ ɫ ɫɨɛɫɬɜɟɧɧɨɣ ɱɚɫɬɨ ɬɨɣ ω = g l Ɉɛɪɚɬɢɦɜɧɢɦɚɧɢɟɧɚɬɨɱɬɨɜ ɜɵɪɚɠɟɧɢɢɞɥɹɮɭɧɤɰɢɢɅɚ
ɝɪɚɧɠɚ ɤɢɧɟɦɚɬɢɱɟɫɤɢɦ ɩɚɪɚɦɟɬɪɨɦ ɹɜɥɹɟɬɫɹ ɜɟɥɢɱɢɧɚ ml ± ɦɨɦɟɧɬ ɢɧɟɪ ɰɢɢ ɱɚɫɬɢɰɵ
ɇɚɣɞɟɦ ɨɛɨɛɳɟɧɧɵɣ ɢɦɩɭɥɶɫ p = ddLϕ = ml ϕ Ɂɚɩɢɲɟɦ ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɮɭɧɤɰɢɢ Ƚɚɦɢɥɶɬɨɧɚ
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Ɍɚɤɢɦ ɨɛɪɚɡɨɦ
p = ml ϕ ɢϕ + gl ϕ = ± ɭɪɚɜɧɟɧɢɟ ɝɚɪɦɨɧɢɱɟɫɤɢɯ ɤɨɥɟɛɚɧɢɣ
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Лекция11.ГАРМОНИЧЕСКИЕКОЛЕБАНИЯМЕХАНИЧЕСКИХСИСТЕМСОДНОЙСТЕПЕНЬЮСВОБОДЫ
ϕ
m
Ɋɢɫ. 11.1.Ɇɚɬɟɦɚɬɢɱɟɫɤɢɣ ɦɚɹɬɧɢɤ
65
ɅȿɄɐɂə ɄɈɅȿȻȺɇɂə ɋɂɋɌȿɆ
ɋ ɇȿɋɄɈɅɖɄɂɆɂ ɋɌȿɉȿɇəɆɂ ɋȼɈȻɈȾɕ
Ʉɨɥɟɛɚɬɟɥɶɧɵɟ ɫɢɫɬɟɦɵ ɫ ɧɟɫɤɨɥɶɤɢɦɢ ɫɬɟɩɟɧɹɦɢ ɫɜɨɛɨɞɵ
Ɋɚɫɫɦɨɬɪɢɦɫɢɫɬɟɦɵɫ ɧɟɫɤɨɥɶɤɢɦɢɫɬɟɩɟɧɹɦɢɫɜɨɛɨɞɵɤɨɬɨɪɵɟɦɨɝɭɬ ɫɨɜɟɪɲɚɬɶ ɤɨɥɟɛɚɬɟɥɶɧɵɟ ɞɜɢɠɟɧɢɹ ɜɛɥɢɡɢ ɭɫɬɨɣɱɢɜɵɯ ɩɨɥɨɠɟɧɢɣ ɪɚɜɧɨɜɟɫɢɹ ɉɪɢɦɟɪɵ ɬɚɤɢɯ ɫɢɫɬɟɦ ɩɨɤɚɡɚɧɵ ɧɚ ɪɢɫ
l l l
ϕ l ϕ ϕ
ϕ
ɚ ɛ
Ɋɢɫ. 12.1.ɋɢɫɬɟɦɵ ɫ ɞɜɭɦɹ ɫɬɟɩɟɧɹɦɢ ɫɜɨɛɨɞɵ
Ⱦɚɧɧɵɟ ɤɨɥɟɛɚɬɟɥɶɧɵɟ ɫɢɫɬɟɦɵ ɢɦɟɸɬ ɞɜɟ ɫɬɟɩɟɧɢ ɫɜɨɛɨɞɵ ɉɪɢ ɡɚɞɚɧ ɧɵɯɡɧɚɱɟɧɢɹɯ l l ɜ ɤɚɱɟɫɬɜɟɨɛɨɛɳɟɧɧɵɯɤɨɨɪɞɢɧɚɬɦɨɠɧɨɜɡɹɬɶɭɝɥɵɨɬ ɤɥɨɧɟɧɢɹϕ ϕ
Ɇɧɨɝɨɚɬɨɦɧɵɟ ɧɟɥɢɧɟɣɧɵɟ ɦɨɥɟɤɭɥɵ ɢɦɟɸɬ N − ɤɨɥɟɛɚɬɟɥɶɧɵɯ ɫɬɟ ɩɟɧɟɣɫɜɨɛɨɞɵɥɢɧɟɣɧɵɟɦɧɨɝɨɚɬɨɦɧɵɟɦɨɥɟɤɭɥɵɢɦɟɸɬ N − ɤɨɥɟɛɚɬɟɥɶ ɧɵɯ ɫɬɟɩɟɧɟɣ ɫɜɨɛɨɞɵ Ɂɞɟɫɶ N ± ɱɢɫɥɨ ɚɬɨɦɨɜ ɜ ɦɨɥɟɤɭɥɟ ɉɪɢɦɟɪɨɦ ɥɢɧɟɣ ɧɨɣ ɬɪɟɯɚɬɨɦɧɨɣ ɦɨɥɟɤɭɥɵ ɦɨɠɟɬ ɫɥɭɠɢɬɶ ɦɨɥɟɤɭɥɚCO ɇɚ ɪɢɫ ɩɨɤɚɡɚ ɧɵ ɧɨɪɦɚɥɶɧɵɟ ɫɦ ɧɢɠɟ ɤɨɥɟɛɚɧɢɹ ɞɚɧɧɨɣ ɦɨɥɟɤɭɥɵ
O C O
Ɋɢɫ. 12.2.ɇɨɪɦɚɥɶɧɵɟ ɤɨɥɟɛɚɧɢɹ ɦɨɥɟɤɭɥɵCO
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Лекция12.КОЛЕБАНИЯСИСТЕМСНЕСКОЛЬКИМИСТЕПЕНЯМИСВОБОДЫ
Ʉɢɧɟɬɢɱɟɫɤɚɹ ɢ ɩɨɬɟɧɰɢɚɥɶɧɚɹ ɷɧɟɪɝɢɹ ɤɨɥɟɛɚɬɟɥɶɧɨɣ ɫɢɫɬɟɦɵ
ɫ ɧɟɫɤɨɥɶɤɢɦɢ ɫɬɟɩɟɧɹɦɢ ɫɜɨɛɨɞɵ Ɏɭɧɤɰɢɹ Ʌɚɝɪɚɧɠɚ
ȼ ɥɟɤɰɢɢ ɛɵɥɨ ɩɨɤɚɡɚɧɨ ɱɬɨ ɤɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɫɢɫɬɟɦɵ ɫɨ ɦɧɨɝɢɦɢ
ɫɬɟɩɟɧɹɦɢ ɫɜɨɛɨɞɵ ɦɨɠɟɬ ɛɵɬɶ ɩɪɟɞɫɬɚɜɥɟɧɚ ɜ ɜɢɞɟ
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Ɂɞɟɫɶ ɮɭɧɤɰɢɢ f ɢT ɹɜɧɨ ɡɚɜɢɫɹɬ ɨɬ ɨɛɨɛɳɟɧɧɵɯ ɤɨɨɪɞɢɧɚɬ ɱɚɫɬɢɰ ɢ ɜɪɟɦɟɧɢ
Ʉɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɫɢɫɬɟɦɵ ɱɚɫɬɢɰ ɹɜɥɹɟɬɫɹ ɧɟɨɞɧɨɪɨɞɧɨɣ ɤɜɚɞɪɚ ɬɢɱɧɨɣ ɮɨɪɦɨɣ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɛɨɛɳɟɧɧɵɯ ɫɤɨɪɨɫɬɟɣ
ȼ ɬɨɦ ɫɥɭɱɚɟ ɤɨɝɞɚ ɧɚ ɫɢɫɬɟɦɭ ɧɚɤɥɚɞɵɜɚɸɬɫɹ ɫɬɚɰɢɨɧɚɪɧɵɟ ɫɜɹɡɢ ɞɥɹ
ɤɢɧɟɬɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ
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ɢ ɮɭɧɤɰɢɹ Ʌɚɝɪɚɧɠɚ ɜ ɫɥɭɱɚɟ ɚɤɬɢɜɧɵɯ ɩɨɬɟɧɰɢɚɥɶɧɵɯ ɫɢɥ ɢɦɟɟɬ ɜɢɞ
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Ʉɨɷɮɮɢɰɢɟɧɬɵ fkl ɧɨɫɹɬ ɧɚɡɜɚɧɢɟɤɨɷɮɮɢɰɢɟɧɬɨɜ ɢɧɟɪɰɢɢȼ ɤɚɱɟɫɬɜɟ ɢɯ
ɦɨɝɭɬ ɜɵɫɬɭɩɚɬɶ ɧɚɩɪɢɦɟɪ ɦɚɫɫɵ ɢ ɦɨɦɟɧɬɵ ɢɧɟɪɰɢɢ ɱɚɫɬɢɰ ɫɢɫɬɟɦɵ Ʉɨɧɤɪɟɬɢɡɢɪɭɟɦ ɬɟɩɟɪɶ ɜɢɞ ɩɨɬɟɧɰɢɚɥɶɧɨɣ ɷɧɟɪɝɢɢ ɜ ɮɨɪɦɭɥɟ
ɫɱɢɬɚɹ ɱɬɨ ɨɧɚ ɧɟ ɡɚɜɢɫɢɬ ɨɬ ɜɪɟɦɟɧɢ Ɋɚɡɥɨɠɢɦ ɩɨɬɟɧɰɢɚɥɶɧɭɸ ɷɧɟɪɝɢɸ ɜ ɪɹɞ ɩɨ ɫɬɟɩɟɧɹɦq q qs ɜɛɥɢɡɢ ɭɫɬɨɣɱɢɜɨɝɨ ɩɨɥɨɠɟɧɢɹ ɪɚɜɧɨɜɟɫɢɹ ɫɢɫɬɟɦɵ
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ɉɟɪɜɨɟ ɫɥɚɝɚɟɦɨɟ ɦɨɠɧɨ ɫɱɢɬɚɬɶ ɪɚɜɧɵɦ ɧɭɥɸ ɩɨɥɚɝɚɹ ɱɬɨ ɩɨɬɟɧɰɢɚɥɶ
ɧɚɹ ɷɧɟɪɝɢɹ ɜ ɩɨɥɨɠɟɧɢɢ ɪɚɜɧɨɜɟɫɢɹ ɪɚɜɧɚ ɧɭɥɸ ɧɨɪɦɢɪɨɜɤɚ ɩɨɬɟɧɰɢɚɥɶɧɨɣ ɷɧɟɪɝɢɢ ȼɫɟ ɤɨɷɮɮɢɰɢɟɧɬɵ ɩɪɢ ɤɨɨɪɞɢɧɚɬɚɯq q qs ɜ ɩɟɪɜɨɣ ɫɬɟɩɟɧɢ ɬɚɤɠɟ
ɪɚɜɧɵ ɧɭɥɸ ɞɥɹ ɩɨɥɨɠɟɧɢɹ ɭɫɬɨɣɱɢɜɨɝɨ ɪɚɜɧɨɜɟɫɢɹ ȼɜɟɞɟɦ ɨɛɨɡɧɚɱɟɧɢɹ
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М. А. Михайлов. ЛЕКЦИИ ПО КЛАССИЧЕСКОЙ МЕХАНИКЕ
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ɗɬɢ ɤɨɷɮɮɢɰɢɟɧɬɵ ɧɚɡɵɜɚɸɬɫɹɨɛɨɛɳɟɧɧɵɦɢ ɤɨɷɮɮɢɰɢɟɧɬɚɦɢ ɠɟɫɬɤɨ ɫɬɢɈɧɢ ɫɢɦɦɟɬɪɢɱɧɵ ɬ ɟ
kkl = klk
ȼ ɪɟɡɭɥɶɬɚɬɟ ɞɥɹ ɩɨɬɟɧɰɢɚɥɶɧɨɣ ɷɧɟɪɝɢɢ ɩɨɥɭɱɚɟɦ
U = ¦¦s s kkl qk ql +
k= l=
ɝɞɟ ɬɪɟɦɹ ɬɨɱɤɚɦɢ ɨɛɨɡɧɚɱɟɧɵ ɱɥɟɧɵ ɜɵɫɲɢɯ ɩɨɪɹɞɤɨɜ Ⱦɥɹ ɫɥɭɱɚɹ ɦɚɥɵɯ ɤɨ ɥɟɛɚɧɢɣɢɫɩɨɥɶɡɭɟɬɫɹɝɚɪɦɨɧɢɱɟɫɤɨɟɩɪɢɛɥɢɠɟɧɢɟ ɜ ɤɨɬɨɪɨɦɜɫɟɦɢɧɟɜɵɩɢ ɫɚɧɧɵɦɢ ɱɥɟɧɚɦɢ ɫɬɚɪɲɢɯ ɩɨɪɹɞɤɨɜ ɩɪɟɧɟɛɪɟɝɚɸɬ ȼ ɪɟɡɭɥɶɬɚɬɟ ɮɭɧɤɰɢɹ Ʌɚ ɝɪɚɧɠɚ ɤɨɥɟɛɚɬɟɥɶɧɨɣ ɫɢɫɬɟɦɵ ɜ ɝɚɪɦɨɧɢɱɟɫɤɨɦ ɩɪɢɛɥɢɠɟɧɢɢ ɢɦɟɟɬ ɜɢɞ
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Ɇɚɥɵɟ ɤɨɥɟɛɚɧɢɹ ɫɢɫɬɟɦɵ ɫ ɞɜɭɦɹ ɫɬɟɩɟɧɹɦɢ ɫɜɨɛɨɞɵ
Ɋɚɫɫɦɨɬɪɢɦ ɫɢɫɬɟɦɭ ɫ ɞɜɭɦɹ ɫɬɟɩɟɧɹɦɢ ɫɜɨɛɨɞɵ ɤɨɬɨɪɚɹ ɫɨɜɟɪɲɚɟɬ ɦɚ
ɥɵɟ ɤɨɥɟɛɚɧɢɹ ɜɛɥɢɡɢ ɭɫɬɨɣɱɢɜɨɝɨ ɩɨɥɨɠɟɧɢɹ ɪɚɜɧɨɜɟɫɢɹ Ɉɫɧɨɜɧɵɟ ɜɵɜɨɞɵ
ɞɚɧɧɨɝɨ ɪɚɫɫɦɨɬɪɟɧɢɹ ɦɨɝɭɬ ɛɵɬɶ ɛɟɡ ɢɡɦɟɧɟɧɢɣ ɪɚɫɩɪɨɫɬɪɚɧɟɧɵ ɧɚ ɫɢɫɬɟɦɵ
ɫ ɩɪɨɢɡɜɨɥɶɧɵɦ ɱɢɫɥɨɦ ɫɬɟɩɟɧɟɣ ɫɜɨɛɨɞɵ
Ɏɭɧɤɰɢɹ Ʌɚɝɪɚɧɠɚ ɞɥɹ ɫɢɫɬɟɦɵ ɫ ɞɜɭɦɹ ɫɬɟɩɟɧɹɦɢ ɫɜɨɛɨɞɵ ɢɦɟɟɬ
ɜɢɞ
L = T − U
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U= (k q + k q q + k q )
Ʉɨɷɮɮɢɰɢɟɧɬɵ ɢɧɟɪɰɢɢ ɜ ɮɨɪɦɭɥɟ ɜ ɨɛɳɟɦ ɫɥɭɱɚɟ ɡɚɜɢɫɹɬ ɨɬ ɤɨ ɨɪɞɢɧɚɬ q q Ɋɚɡɥɨɠɢɦ ɢɯ ɜ ɪɹɞ ɩɨ ɫɬɟɩɟɧɹɦ q q ɢ ɨɝɪɚɧɢɱɢɦɫɹ ɱɥɟɧɚɦɢ ɩɟɪɜɨɝɨ ɩɨɪɹɞɤɚ
f (q q ) f ( )= f
Ɍɨɝɞɚ ɤɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ
T= (f q + f q q + f q )
Ɂɞɟɫɶ ɜɫɟ ɤɨɷɮɮɢɰɢɟɧɬɵ ɢɧɟɪɰɢɢ ± ɩɨɫɬɨɹɧɧɵɟ ɱɢɫɥɚ
Ⱦɥɹ ɩɨɥɨɠɟɧɢɹ ɭɫɬɨɣɱɢɜɨɝɨ ɪɚɜɧɨɜɟɫɢɹ ɤɜɚɞɪɚɬɢɱɧɚɹ ɮɨɪɦɚ ɩɨɥɨ
ɠɢɬɟɥɶɧɚ ɤɚɤ ɢ ɤɜɚɞɪɚɬɢɱɧɚɹ ɮɨɪɦɚ ɤɢɧɟɬɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ Ʉɨɷɮɮɢɰɢ
ɟɧɬɵ ɢɧɟɪɰɢɢ ɢ ɤɨɷɮɮɢɰɢɟɧɬɵ ɠɟɫɬɤɨɫɬɢ ɫɥɟɞɨɜɚɬɟɥɶɧɨ ɭɞɨɜɥɟɬɜɨɪɹɸɬ ɤɪɢ
ɬɟɪɢɸ ɋɢɥɶɜɟɫɬɪɚ
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Лекция12.КОЛЕБАНИЯСИСТЕМСНЕСКОЛЬКИМИСТЕПЕНЯМИСВОБОДЫ
f > f f − f >
k > k k − k >
Ⱦɢɮɮɟɪɟɧɰɢɚɥɶɧɵɟ ɭɪɚɜɧɟɧɢɹ ɤɨɥɟɛɚɧɢɣ ɩɨɥɭɱɚɸɬɫɹ ɢɡ ɭɪɚɜɧɟɧɢɣ Ʌɚ
ɝɪɚɧɠɚ
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ɢ ɜɵɪɚɠɟɧɢɣ ɢ |
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ɝɞɟ ɢɫɩɨɥɶɡɨɜɚɧɵ ɪɚɜɟɧɫɬɜɚ f |
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ɋɥɟɞɨɜɚɬɟɥɶɧɨ ɦɚɥɵɟ ɤɨɥɟɛɚɧɢɹ ɫɢɫɬɟɦɵ ɫ ɞɜɭɦɹ ɫɬɟɩɟɧɹɦɢ ɫɜɨɛɨɞɵ |
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ɨɤɨɥɨ ɩɨɥɨɠɟɧɢɹ ɭɫɬɨɣɱɢɜɨɝɨ ɪɚɜɧɨɜɟɫɢɹ ɨɩɢɫɵɜɚɸɬɫɹ ɞɜɭɦɹ ɨɞɧɨɪɨɞɧɵɦɢ ɥɢɧɟɣɧɵɦɢ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɦɢ ɭɪɚɜɧɟɧɢɹɦɢ ɜɬɨɪɨɝɨ ɩɨɪɹɞɤɚ ɫ ɩɨɫɬɨɹɧɧɵɦɢ ɤɨɷɮɮɢɰɢɟɧɬɚɦɢ
Ɋɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɣ ɢɳɟɦ ɜ ɜɢɞɟ
q = AVLQ(ωt + α )
®¯q = BVLQ(ωt + α )
ɝɞɟA B ω α ± ɧɟɢɡɜɟɫɬɧɵɟ ɩɨɫɬɨɹɧɧɵɟ Ⱦɢɮɮɟɪɟɧɰɢɪɭɹ ɞɜɚɠɞɵ ɩɨ ɜɪɟɦɟɧɢ ɭɪɚɜɧɟɧɢɹ ɩɨɞɫɬɚɜɥɹɹ ɩɨɥɭ
ɱɟɧɧɵɟ ɪɟɡɭɥɶɬɚɬɵ ɜ ɭɪɚɜɧɟɧɢɹ ɩɨɥɭɱɢɦ
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ɋɢɫɬɟɦɚ ɫɨɞɟɪɠɢɬ ɚɥɝɟɛɪɚɢɱɟɫɤɢɟ ɨɞɧɨɪɨɞɧɵɟ ɥɢɧɟɣɧɵɟ ɭɪɚɜɧɟ ɧɢɹ ɞɥɹ ɧɚɯɨɠɞɟɧɢɹ ɚɦɩɥɢɬɭɞ A B ɋɢɫɬɟɦɚ ɭɪɚɜɧɟɧɢɣ ɢɦɟɟɬ ɪɟɲɟɧɢɟ ɨɬɥɢɱɧɨɟ ɨɬ ɧɭɥɹ ɟɫɥɢ ɨɩɪɟɞɟɥɢɬɟɥɶ ɷɬɨɣ ɫɢɫɬɟɦɵ ɪɚɜɟɧ ɧɭɥɸ
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k − f ω |
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ɍɪɚɜɧɟɧɢɟ ɧɚɡɵɜɚɟɬɫɹɜɟɤɨɜɵɦɢɥɢɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɢɦ ɭɪɚɜɧɟ ɧɢɟɦ ɱɚɫɬɨɬɈɧɨ ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɤɜɚɞɪɚɬɧɨɟ ɭɪɚɜɧɟɧɢɟ ɨɬɧɨɫɢɬɟɥɶɧɨω Ʉɨɪɧɢ ɟɝɨ ɜɟɳɟɫɬɜɟɧɧɵ ɢ ɩɨɥɨɠɢɬɟɥɶɧɵ Ɉɛɨɡɧɚɱɢɦ ɢɯω ω ɂɡɜɥɟɱɟɦ ɤɨɪɧɢ ɢ ɢɫɩɨɥɶɡɭɟɦ ɩɨɥɨɠɢɬɟɥɶɧɨɫɬɶ ɡɧɚɱɟɧɢɣ ω ω Ⱦɚɧɧɵɟ ɱɚɫɬɨɬɵ ɧɚɡɵɜɚɸɬɫɹ ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɢɦɢ ɱɚɫɬɨɬɚɦɢɄɚɠɞɨɣ ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɣ ɱɚɫɬɨɬɟ ɨɬɜɟɱɚɟɬ ɨɞɧɨ ɱɚɫɬɧɨɟ ɪɟɲɟɧɢɟ Ʉɚɠɞɨɣ ɱɚɫɬɨɬɟ ɨɬɜɟɱɚɸɬ ɫɜɨɢ ɡɧɚɱɟɧɢɹ A B α ɑɚɫɬɧɵɟ ɪɟɲɟɧɢɹ ɥɢɧɟɣɧɨ ɧɟɡɚɜɢɫɢɦɵ ɩɨɷɬɨɦɭ ɨɛɳɟɟ ɪɟɲɟɧɢɟ ɟɫɬɶ ɢɯ ɥɢɧɟɣ ɧɚɹ ɤɨɦɛɢɧɚɰɢɹ
q = A VLQ(ω t +α )+ A VLQ(ω t +α ) q = B VLQ(ω t +α )+ B VLQ(ω t +α )
Ɇɟɠɞɭ ɱɢɫɥɚɦɢ A ɢB A ɢB ɢɦɟɟɬɫɹ ɫɜɹɡɶ Ɉɧɚ ɭɫɬɚɧɚɜɥɢɜɚɟɬɫɹ ɭɪɚɜɧɟɧɢɹɦɢ ɗɬɚ ɫɜɹɡɶ ɫɥɟɞɭɸɳɚɹ
69
М. А. Михайлов. ЛЕКЦИИ ПО КЛАССИЧЕСКОЙ МЕХАНИКЕ
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ɂ ɨɤɨɧɱɚɬɟɥɶɧɨ ɨɛɳɟɟ ɪɟɲɟɧɢɟ |
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ȼɨɤɨɧɱɚɬɟɥɶɧɨɦɪɟɲɟɧɢɢɱɚɫɬɨɬɵ ω ω ɢ ɤɨɷɮɮɢɰɢɟɧɬɵ μ μ ±ɧɚɣ ɞɟɧɧɵɟ ɱɢɫɥɚ ɤɨɬɨɪɵɟ ɯɚɪɚɤɬɟɪɢɡɭɸɬ ɤɨɥɟɛɚɬɟɥɶɧɭɸ ɫɢɫɬɟɦɭ ɜɟɥɢɱɢɧɵ A A α α ± ɩɪɨɢɡɜɨɥɶɧɵɟ ɩɨɫɬɨɹɧɧɵɟ ɤɨɬɨɪɵɟ ɨɩɪɟɞɟɥɹɸɬɫɹ ɧɚɱɚɥɶɧɵɦɢ ɭɫɥɨɜɢɹɦɢ
ɂɡ ɮɨɪɦɭɥ ɜɢɞɧɨ ɱɬɨ ɤɨɥɟɛɚɬɟɥɶɧɨɟ ɞɜɢɠɟɧɢɟ ɫɢɫɬɟɦɵ ɫ ɞɜɭɦɹ
ɫɬɟɩɟɧɹɦɢ ɫɜɨɛɨɞɵ ɨɤɨɥɨ ɭɫɬɨɣɱɢɜɨɝɨ ɩɨɥɨɠɟɧɢɹ ɪɚɜɧɨɜɟɫɢɹ ɫɤɥɚɞɵɜɚɟɬɫɹ ɢɡ
ɞɜɭɯ ɧɟɡɚɜɢɫɢɦɵɯ ɤɨɥɟɛɚɧɢɣ
Q = A VLQ(ω t +α ) Q = A VLQ(ω t +α )
ɉɟɪɜɨɟ ɤɨɥɟɛɚɧɢɟ ɩɪɨɢɫɯɨɞɢɬ ɫ ɱɚɫɬɨɬɨɣ ω ɜɬɨɪɨɟ ɤɨɥɟɛɚɧɢɟ ± ɫ ɱɚɫɬɨɬɨɣω ɗɬɢ ɤɨɥɟɛɚɧɢɹ ɧɚɡɵɜɚɸɬɫɹɧɨɪɦɚɥɶɧɵɦɢ ɤɨɥɟɛɚɧɢɹɦɢ
ȼ ɤɚɠɞɨɦ ɧɨɪɦɚɥɶɧɨɦ ɤɨɥɟɛɚɧɢɢ ɜɫɟ ɤɨɨɪɞɢɧɚɬɵ ɢɡɦɟɧɹɸɬɫɹ ɩɨ ɝɚɪɦɨ
ɧɢɱɟɫɤɨɦɭ ɡɚɤɨɧɭ ɢɦɟɹ ɨɞɢɧɚɤɨɜɵɟ ɱɚɫɬɨɬɵ ɢ ɮɚɡɵ Ɉɧɢ ɨɞɧɨɜɪɟɦɟɧɧɨ ɨɛɪɚ
ɳɚɸɬɫɹ ɜ ɧɭɥɶ ɨɞɧɨɜɪɟɦɟɧɧɨ ɞɨɫɬɢɝɚɸɬ ɦɚɤɫɢɦɚɥɶɧɵɯ ɡɧɚɱɟɧɢɣ Ⱥɦɩɥɢɬɭɞɵ
ɢɡɦɟɧɟɧɢɹ ɤɨɨɪɞɢɧɚɬ ɜ ɤɚɠɞɨɦ ɧɨɪɦɚɥɶɧɨɦ ɤɨɥɟɛɚɧɢɢ ɧɚɯɨɞɹɬɫɹ ɜ ɩɨɫɬɨɹɧɧɨɦ ɨɬɧɨɲɟɧɢɢμ ± ɞɥɹ ɩɟɪɜɨɝɨ ɤɨɥɟɛɚɧɢɹμ ± ɞɥɹ ɜɬɨɪɨɝɨ ɧɨɪɦɚɥɶɧɨɝɨ ɤɨɥɟɛɚ ɧɢɹ ɧɟ ɡɚɜɢɫɹɳɟɦ ɨɬ ɧɚɱɚɥɶɧɵɯ ɭɫɥɨɜɢɣ
ȼ ɫɥɭɱɚɟ ɤɨɝɞɚ ɫɢɫɬɟɦɚ ɢɦɟɟɬ s ɫɬɟɩɟɧɟɣ ɫɜɨɛɨɞɵ ɤɨɥɟɛɚɬɟɥɶɧɨɟ ɞɜɢ ɠɟɧɢɟ ɪɚɫɩɚɞɚɟɬɫɹ ɧɚs ɧɨɪɦɚɥɶɧɵɯ ɤɨɥɟɛɚɧɢɣ ɤɚɠɞɨɟ ɢɡ ɤɨɬɨɪɵɯ ɯɚɪɚɤɬɟɪɢ ɡɭɟɬɫɹ ɫɜɨɟɣ ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɣ ɱɚɫɬɨɬɨɣ ȼ ɱɚɫɬɧɨɦ ɫɥɭɱɚɟ ɧɟɤɨɬɨɪɵɟ ɱɚɫɬɨ ɬɵ ɦɨɝɭɬ ɫɨɜɩɚɞɚɬɶ ɞɪɭɝ ɫ ɞɪɭɝɨɦ ɇɚ ɪɢɫ ɩɨɤɚɡɚɧɵ ɧɨɪɦɚɥɶɧɵɟ ɤɨɥɟɛɚ ɧɢɹɦɨɥɟɤɭɥɵ CO ȼɫɟɝɨɢɦɟɟɬɫɹɱɟɬɵɪɟɧɨɪɦɚɥɶɧɵɯɤɨɥɟɛɚɧɢɹɅɢɧɟɣɧɵɟ ɤɨɥɟɛɚɧɢɹ ɦɨɥɟɤɭɥɵ ɯɚɪɚɤɬɟɪɢɡɭɸɬɫɹ ɱɚɫɬɨɬɚɦɢω ω ɑɚɫɬɨɬɵ ɞɜɭɯ ɧɟɥɢɧɟɣ ɧɵɯ ɤɨɥɟɛɚɧɢɣ ɫɨɜɩɚɞɚɸɬ ɞɪɭɝ ɫ ɞɪɭɝɨɦω = ω
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