Лекции по классической механике. Учебное пособие
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Лекция7.ЗАДАЧАКЕПЛЕРА.РАССЕЯНИЕЧАСТИЦ
ɬɪɚɟɤɬɨɪɢɢ ɞɜɢɠɟɧɢɹ ɱɚɫɬɢɰɵ Ⱦɥɹ ɧɚɯɨɠɞɟɧɢɹ ɬɪɚɟɤɬɨɪɢɢ ɞɜɢɠɟɧɢɹ ɱɚɫɬɢɰɵ
ɦɨɠɧɨ ɜɨɫɩɨɥɶɡɨɜɚɬɶɫɹ ɭɪɚɜɧɟɧɢɟɦ Ȼɢɧɟ Ⱦɥɹ ɷɬɨɝɨ ɫɥɟɞɭɟɬ ɩɨ ɮɨɪɦɭɥɟ |
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ɜɵɱɢɫɥɢɬɶ ɦɨɞɭɥɶ ɫɢɥɵ ɞɟɣɫɬɜɭɸɳɟɣ ɧɚ ɱɚɫɬɢɰɭ
F r = rα
ɢɩɨɞɫɬɚɜɢɬɶɞɚɧɧɨɟɜɵɪɚɠɟɧɢɟɜ ɭɪɚɜɧɟɧɢɟȻɢɧɟɊɟɲɢɜɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ ɧɚɣɞɟɦ ɬɪɚɟɤɬɨɪɢɸ ɞɜɢɠɟɧɢɹ ɱɚɫɬɢɰɵ
Ɋɟɡɭɥɶɬɚɬ ɜɵɱɢɫɥɟɧɢɣ ɬɚɤɨɜ
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ɟɫɬɶ ɮɨɤɚɥɶɧɵɣ ɩɚɪɚɦɟɬɪ ɢ ɷɤɫɰɟɧɬɪɢɫɢɬɟɬ ɬɪɚɟɤɬɨɪɢɢ ɞɜɢɠɟɧɢɹ ɍɪɚɜɧɟɧɢɟ
ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɭɪɚɜɧɟɧɢɟ ɤɪɢɜɨɣ ɜɬɨɪɨɝɨ ɩɨɪɹɞɤɚ ɝɢɩɟɪɛɨɥɚ ɩɚɪɚɛɨɥɚ ɷɥ
ɥɢɩɫ ɨɤɪɭɠɧɨɫɬɶ ɜ ɩɨɥɹɪɧɨɣ ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ ɧɚɱɚɥɨɦ ɤɨɬɨɪɨɣ ɹɜɥɹɟɬɫɹ ɬɨɱɤɚ
ɝɞɟ ɪɚɡɦɟɳɚɟɬɫɹ ɫɢɥɨɜɨɣ ɰɟɧɬɪ ȼɢɞ ɤɪɢɜɨɣ ɜɬɨɪɨɝɨ ɩɨɪɹɞɤɚ ɨɩɪɟɞɟɥɹɟɬɫɹ ɷɤɫɰɟɧ
ɬɪɢɫɢɬɟɬɨɦ
ε > E > ± ɝɢɩɟɪɛɨɥɚ
ε = E = ± ɩɚɪɚɛɨɥɚ
< ε < E < ± ɷɥɥɢɩɫ
ε = E = − mα 
l ± ɨɤɪɭɠɧɨɫɬɶ
ɗɥɥɢɩɬɢɱɟɫɤɚɹ ɬɪɚɟɤɬɨɪɢɹ ɞɜɢɠɟɧɢɹ ɱɚɫɬɢɰɵ ɩɨɤɚɡɚɧɚ ɧɚ ɪɢɫ Ɍɚɦ
ɠɟ ɢɡɨɛɪɚɠɟɧ ɮɨɤɚɥɶɧɵɣ ɩɚɪɚɦɟɬɪ ɤɪɢɜɨɣ
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S Μ
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Ɋɢɫ. 7.2.ɗɥɥɢɩɬɢɱɟɫɤɚɹ ɬɪɚɟɤɬɨɪɢɹ ɞɜɢɠɟɧɢɹ ɱɚɫɬɢɰɵ
41
М. А. Михайлов. ЛЕКЦИИ ПО КЛАССИЧЕСКОЙ МЕХАНИКЕ
Ɋɚɫɫɟɹɧɢɟ ɱɚɫɬɢɰɵ ɧɚ ɫɢɥɨɜɨɦ ɰɟɧɬɪɟ
Ɋɚɫɫɦɨɬɪɢɦ ɡɚɞɚɱɭ ɨ ɪɚɫɫɟɹɧɢɢ ɱɚɫɬɢɰɵ ɧɚ ɫɢɥɨɜɨɦ ɰɟɧɬɪɟ Ȼɭɞɟɦ ɪɚɫ
ɫɦɚɬɪɢɜɚɬɶ ɞɜɢɠɟɧɢɟ ɱɚɫɬɢɰɵ ɜ ɫɢɥɨɜɨɦ ɩɨɥɟ
U r = αr α >
ɉɪɢɦɟɪɨɦ ɞɚɧɧɨɣ ɡɚɞɚɱɢ ɫɥɭɠɢɬ ɡɚɞɚɱɚ ɨ ɞɜɢɠɟɧɢɢ ɱɚɫɬɢɰɵ ɜ ɤɭɥɨɧɨɜɫɤɨɦ ɩɨɥɟ ɨɬɬɚɥɤɢɜɚɧɢɹ Ƚɪɚɮɢɤ ɷɮɮɟɤɬɢɜɧɨɝɨ ɩɨɬɟɧɰɢɚɥɚ ɞɥɹ ɱɚɫ ɬɢɰɵ ɜ ɬɚɤɨɦ ɩɨɥɟ ɩɪɟɞɫɬɚɜɥɟɧ ɧɚ ɪɢɫ
U ɷɮɮ r
E
r r
Ɋɢɫ. 7.3.ɗɮɮɟɤɬɢɜɧɵɣ ɩɨɬɟɧɰɢɚɥ ɱɚɫɬɢɰɵ ɜ ɤɭɥɨɧɨɜɫɤɨɦ ɩɨɥɟ ɨɬɬɚɥɤɢɜɚɧɢɹ
ɋ ɋ ɋ
Ɋɢɫ. 7.4.Ɍɪɚɟɤɬɨɪɢɹ ɜ ɩɨɥɟ Ɋɢɫ. 7.5.Ɍɪɚɟɤɬɨɪɢɹ ɜ ɩɨɥɟ ɩɪɢɬɹɠɟɧɢɹ ɨɬɬɚɥɤɢɜɚɧɢɹ
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Лекция7.ЗАДАЧАКЕПЛЕРА.РАССЕЯНИЕЧАСТИЦ
Ⱦɜɢɠɟɧɢɟ ɱɚɫɬɢɰɵ ɜɨɡɦɨɠɧɨ ɩɪɢ ɡɧɚɱɟɧɢɹɯ ɟɟ ɩɨɥɧɨɣ ɷɧɟɪɝɢɢ E > ɋɭɳɟɫɬɜɭɟɬ ɦɢɧɢɦɚɥɶɧɨɟ ɪɚɫɫɬɨɹɧɢɟ ɧɚ ɤɨɬɨɪɨɟ ɱɚɫɬɢɰɚ ɦɨɠɟɬ ɩɨɞɨɣɬɢ ɤ ɫɢɥɨɜɨɦɭ ɰɟɧɬɪɭ ±r ɞɜɢɠɟɧɢɟ ɱɚɫɬɢɰɵ ɢɧɮɢɧɢɬɧɨɟ
Ɂɚɞɚɱɚ ɨ ɧɚɯɨɠɞɟɧɢɢ ɬɪɚɟɤɬɨɪɢɢ ɞɜɢɠɟɧɢɹ ɱɚɫɬɢɰɵ ɪɟɲɚɟɬɫɹ ɚɧɚɥɨɝɢɱɧɨ
ɬɨɦɭ ɤɚɤ ɷɬɨ ɞɟɥɚɥɨɫɶ ɜ ɡɚɞɚɱɟ Ʉɟɩɥɟɪɚ Ɋɟɡɭɥɶɬɚɬ ɫɥɟɞɭɸɳɢɣ
r = |
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ɝɞɟ ɮɨɤɚɥɶɧɵɣ ɩɚɪɚɦɟɬɪ ɢ ɷɤɫɰɟɧɬɪɢɫɢɬɟɬ ɨɩɪɟɞɟɥɹɸɬɫɹ ɮɨɪɦɭɥɚɦɢ Ɍɪɚɟɤɬɨɪɢɟɣ ɞɜɢɠɟɧɢɹ ɱɚɫɬɢɰɵ ɹɜɥɹɟɬɫɹ ɝɢɩɟɪɛɨɥɚ Ƚɢɩɟɪɛɨɥɢɱɟɫɤɢɟ
ɬɪɚɟɤɬɨɪɢɢɞɜɢɠɟɧɢɹɱɚɫɬɢɰɵɜ ɡɚɞɚɱɟɄɟɩɥɟɪɚɤɭɥɨɧɨɜɫɤɨɟɢɥɢɝɪɚɜɢɬɚɰɢ ɨɧɧɨɟ ɩɨɥɟ ɩɪɢɬɹɠɟɧɢɹ ɢ ɜ ɞɚɧɧɨɣ ɡɚɞɚɱɟ ɤɭɥɨɧɨɜɫɤɨɟ ɩɨɥɟ ɪɚɫɫɟɹɧɢɹ ɩɪɟɞ ɫɬɚɜɥɟɧɵ ɧɚ ɪɢɫ ɢ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ ɱɟɪɟɡ ɋ ɨɛɨɡɧɚɱɟɧ ɫɢɥɨɜɨɣ ɰɟɧɬɪ
Ɋɚɫɫɟɹɧɢɟ ɩɭɱɤɚ ɱɚɫɬɢɰ ɧɚ ɚɬɨɦɚɯ ɦɢɲɟɧɢ Ɏɨɪɦɭɥɚ Ɋɟɡɟɪɮɨɪɞɚ
ɉɭɫɬɶ ɩɭɱɨɤ ɱɚɫɬɢɰ ɪɚɫɫɟɢɜɚɟɬɫɹ ɧɚ ɚɬɨɦɚɯ ɦɢɲɟɧɢ ɧɚɩɪɢɦɟɪ ɦɟɬɚɥɥɢ ɱɟɫɤɨɣ ɮɨɥɶɝɟ ɢɧɬɟɧɫɢɜɧɨɫɬɶ ɩɭɱɤɚ I ± ɱɢɫɥɨ ɱɚɫɬɢɰ ɩɪɢɯɨɞɹɳɢɯɫɹ ɧɚ ɟɞɢ ɧɢɰɭ ɩɥɨɳɚɞɢ ɩɨɩɟɪɟɱɧɨɝɨ ɫɟɱɟɧɢɹ ɩɭɱɤɚ ɜ ɟɞɢɧɢɰɭ ɜɪɟɦɟɧɢρ ± ɩɪɢɰɟɥɶɧɵɣ ɩɚɪɚɦɟɬɪ ɪɢɫ
Ⱦɢɮɮɟɪɟɧɰɢɚɥɶɧɵɦ ɫɟɱɟɧɢɟɦ ɪɚɫɫɟɹɧɢɹ dσ Ω 
dΩ ɜ ɫɢɫɬɟɦɟ ɰɟɧɬɪɚ ɢɧɟɪɰɢɢ ɧɚɡɵɜɚɟɬɫɹ ɨɬɧɨɲɟɧɢɟ ɱɢɫɥɚ ɱɚɫɬɢɰ ɪɚɫɫɟɢɜɚɟɦɵɯ ɜ ɟɞɢɧɢɰɭ ɜɪɟɦɟɧɢ ɜ ɢɧɬɟɪɜɚɥɟ ɬɟɥɟɫɧɨɝɨ ɭɝɥɚ ɨɬΩ ɞɨΩ + dΩ ɤ ɢɧɬɟɧɫɢɜɧɨɫɬɢ ɩɚɞɚɸɳɟɝɨ ɩɭɱɤɚ ɉɨɫɤɨɥɶɤɭdΩ = π VLQθ dθ ɢ ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ
π Iρ dρ = π I(dσ 
dΩ)VLQθ dθ
ɞɥɹ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɫɟɱɟɧɢɹ ɪɚɫɫɟɹɧɢɹ ɩɨɥɭɱɚɟɦ
dσ θ |
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m v&∞ ϕ θ
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ρ r&∞
ɋ
Ɋɢɫ. 7.6.Ɋɚɫɫɟɹɧɢɟ ɱɚɫɬɢɰɵ ɧɚ ɫɢɥɨɜɨɦ ɰɟɧɬɪɟ
43
М. А. Михайлов. ЛЕКЦИИ ПО КЛАССИЧЕСКОЙ МЕХАНИКЕ
Ⱦɥɹɜɵɱɢɫɥɟɧɢɹɞɚɧɧɨɣɮɨɪɦɭɥɵɧɟɨɛɯɨɞɢɦɨɡɧɚɬɶɡɚɜɢɫɢɦɨɫɬɶ ρ ɨɬ ɭɝɥɚ θ ɬ ɟ ɜɵɱɢɫɥɟɧɢɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɫɟɱɟɧɢɹ ɪɚɫɫɟɹɧɢɹ ɫɜɨɞɢɬɫɹ ɤ ɧɚɯɨɠɞɟɧɢɸ ɮɭɧɤɰɢɨɧɚɥɶɧɨɣɡɚɜɢɫɢɦɨɫɬɢɩɪɢɰɟɥɶɧɨɝɨɩɚɪɚɦɟɬɪɚ ɨɬɭɝɥɚ ɪɚɫɫɟɹɧɢɹ
ɍɝɨɥ ɪɚɫɫɟɹɧɢɹ ɩɪɟɞɫɬɚɜɢɦ ɜ ɜɢɞɟ
θ = π − ϕ
ɍɝɨɥϕ ɨɩɪɟɞɟɥɢɦ ɩɨ ɮɨɪɦɭɥɟ
∞ |
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ɂɡ ɞɜɭɯ ɩɨɫɥɟɞɧɢɯ ɮɨɪɦɭɥ ɜɢɞɧɨ ɱɬɨ ɭɝɨɥ ɪɚɫɫɟɹɧɢɹθ ɹɜɥɹɟɬɫɹ ɮɭɧɤɰɢ ɟɣ ɩɨɥɧɨɣ ɷɧɟɪɝɢɢ E ɢ ɦɨɦɟɧɬɚ ɢɦɩɭɥɶɫɚl ɱɚɫɬɢɰɵm Ⱦɥɹ ɧɚɯɨɠɞɟɧɢɹ ɡɚɜɢ ɫɢɦɨɫɬɢρ θ ɜɵɪɚɡɢɦ ɦɨɦɟɧɬ ɢɦɩɭɥɶɫɚ ɱɚɫɬɢɰɵ ɱɟɪɟɡ ɩɪɢɰɟɥɶɧɵɣ ɩɚɪɚɦɟɬɪ
l = mv∞ρ
ɢ ɢɫɩɨɥɶɡɭɟɦ
E = |
mv∞ |
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ɉɨɞɫɬɚɜɢɜ ɞɚɧɧɵɟ ɮɨɪɦɭɥɵ ɜ ɩɨɥɭɱɢɦ
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ɉɨ ɡɚɞɚɧɧɨɣ ɮɭɧɤɰɢɢU r ɨɩɪɟɞɟɥɹɟɬɫɹ ɡɚɜɢɫɢɦɨɫɬɶ ρ θ ȼ ɩɨɥɹɯ ɨɬ
ɬɚɥɤɢɜɚɧɢɹ ɞɚɧɧɚɹ ɮɨɪɦɭɥɚ ɞɚɟɬ ɨɞɧɨɡɧɚɱɧɭɸ ɡɚɜɢɫɢɦɨɫɬɶ ɩɪɢɰɟɥɶɧɨɝɨ ɩɚɪɚ
ɦɟɬɪɚ ɨɬ ɭɝɥɚ ɪɚɫɫɟɹɧɢɹ Ɂɚɦɟɬɢɦ ɱɬɨ ɞɥɹ ɰɟɧɬɪɚɥɶɧɵɯ ɩɨɥɟɣ ɩɪɢɬɹɠɟɧɢɹ ɫɥɟ ɞɭɟɬ ɜ ɥɟɜɨɣ ɱɚɫɬɢ ɮɨɪɦɭɥɵ ɛɪɚɬɶ ɡɧɚɤ ɩɥɸɫ ɡɚɜɢɫɢɦɨɫɬɶρ θ ɩɪɢ ɷɬɨɦ ɦɨɠɟɬ
ɛɵɬɶ ɧɟɨɞɧɨɡɧɚɱɧɨɣ ɩɪɢɦɟɪ ɩɚɞɟɧɢɟ ɱɚɫɬɢɰɵ ɧɚ ɰɟɧɬɪ ɩɨ ɬɪɚɟɤɬɨɪɢɢ ɜ ɜɢɞɟ
ɡɚɤɪɭɱɢɜɚɸɳɟɣɫɹ ɫɩɢɪɚɥɢ
Ⱦɥɹ ɤɭɥɨɧɨɜɫɤɨɝɨ ɩɨɥɹ ɨɬɬɚɥɤɢɜɚɧɢɹ ɮɭɧɤɰɢɹU r = α r α > ɉɨɞɫɬɚɜɢɜ
ɞɚɧɧɭɸ ɮɭɧɤɰɢɸ ɜ ɮɨɪɦɭɥɭ ɢ ɩɪɨɜɟɞɹ ɜɵɱɢɫɥɟɧɢɟ ɧɚɣɞɟɦ
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Лекция7.ЗАДАЧАКЕПЛЕРА.РАССЕЯНИЕЧАСТИЦ
ɉɪɨɞɢɮɮɟɪɟɧɰɢɪɨɜɚɜ ɞɚɧɧɨɟ ɜɵɪɚɠɟɧɢɟ ɩɨ θ ɩɨɞɫɬɚɜɢɜ ɪɟɡɭɥɶɬɚɬ ɜ ɩɨɥɭɱɢɦ ɢɫɤɨɦɭɸ ɮɨɪɦɭɥɭ Ɋɟɡɟɪɮɨɪɞɚ
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VLQ θ |
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Ⱦɚɧɧɚɹ ɮɨɪɦɭɥɚ ɩɨɤɚɡɵɜɚɟɬ ɱɬɨ ɩɨɞɚɜɥɹɸɳɟɟ ɛɨɥɶɲɢɧɫɬɜɨ ɱɚɫɬɢɰ ɪɚɫ
ɫɟɢɜɚɟɬɫɹ ɧɚ ɦɚɥɵɟ ɭɝɥɵ ɢ ɦɨɠɟɬ ɛɵɬɶ ɢɫɩɨɥɶɡɨɜɚɧɚ ɞɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɪɚɡɦɟɪɨɜ ɚɬɨɦɧɵɯ ɹɞɟɪ
45
ɈɋɇɈȼɕ ȺɇȺɅɂɌɂɑȿɋɄɈɃ ɆȿɏȺɇɂɄɂ
ɅȿɄɐɂə ɈɋɇɈȼɇɕȿ ɉɈɇəɌɂə
ɂ ɌȿɈɊȿɆɕ ȺɇȺɅɂɌɂɑȿɋɄɈɃ ɆȿɏȺɇɂɄɂ
Ɇɟɯɚɧɢɱɟɫɤɚɹ ɫɢɫɬɟɦɚ ɫ ɧɚɥɨɠɟɧɧɵɦɢ ɧɚ ɧɟɟ ɫɜɹɡɹɦɢ
Ʉɥɚɫɫɢɮɢɤɚɰɢɹ ɫɜɹɡɟɣ
Ⱥɧɚɥɢɬɢɱɟɫɤɚɹ ɦɟɯɚɧɢɤɚɢɡɭɱɚɟɬ ɞɜɢɠɟɧɢɟ ɧɟɫɜɨɛɨɞɧɵɯ ɫɢɫɬɟɦ ɱɚɫɬɢɰ ɬ ɟɫɢɫɬɟɦɧɚ ɤɨɬɨɪɵɟɧɚɤɥɚɞɵɜɚɸɬɫɹɤɚɤɢɟ ɬɨɨɝɪɚɧɢɱɟɧɢɹɈɝɪɚɧɢɱɟɧɢɹ ɧɚɤɥɚɞɵɜɚɸɬɫɹ ɧɚ ɤɨɨɪɞɢɧɚɬɵ ɢ ɫɤɨɪɨɫɬɢ ɞɜɢɠɟɧɢɹ ɱɚɫɬɢɰ ɢ ɧɚɡɵɜɚɸɬɫɹɫɜɹ ɡɹɦɢ ɉɪɢɦɟɪɚɦɢɬɚɤɢɯɫɢɫɬɟɦɦɨɝɭɬɫɥɭɠɢɬɶɦɚɬɟɦɚɬɢɱɟɫɤɢɣɢ ɩɪɭɠɢɧɧɵɣ ɦɚɹɬɧɢɤ ɱɚɫɬɢɰɚ ɧɚ ɩɥɨɫɤɨɫɬɢ ɢ ɞɪ
Ⱥɧɚɥɢɬɢɱɟɫɤɢ ɫɜɹɡɢ ɡɚɞɚɸɬɫɹ ɜ ɜɢɞɟ ɭɪɚɜɧɟɧɢɣ ɢɥɢ ɧɟɪɚɜɟɧɫɬɜ ɤɨɬɨɪɵɟ ɫɜɹɡɵɜɚɸɬ ɤɨɨɪɞɢɧɚɬɵ ɢ ɢɥɢ ɫɤɨɪɨɫɬɢ ɱɚɫɬɢɰ ɫɢɫɬɟɦɵ ɞɪɭɝ ɫ ɞɪɭɝɨɦ ȿɫɥɢ ɫɜɹɡɶ ɡɚɞɚɟɬɫɹ ɜ ɜɢɞɟ ɭɪɚɜɧɟɧɢɹ ɬɨ ɨɧɚ ɧɚɡɵɜɚɟɬɫɹɭɞɟɪɠɢɜɚɸɳɟɣɟɫɥɢ ɜ ɜɢɞɟ ɧɟɪɚɜɟɧɫɬɜɚ ±ɧɟɭɞɟɪɠɢɜɚɸɳɟɣȼ ɞɚɥɶɧɟɣɲɟɦ ɦɵ ɛɭɞɟɦ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɬɨɥɶɤɨ ɭɞɟɪɠɢɜɚɸɳɢɟ ɫɜɹɡɢ ɇɚɩɪɢɦɟɪ ɟɫɥɢ ɱɚɫɬɢɰɚ ɦɨɠɟɬ ɞɜɢɝɚɬɶɫɹ ɬɨɥɶɤɨ ɩɨ ɩɨ
ɜɟɪɯɧɨɫɬɢ ɫɮɟɪɵ ɪɚɞɢɭɫɚ |
R ɬɨ ɭɪɚɜɧɟɧɢɟ ɫɜɹɡɢ ɟɫɬɶ x + y + z − R = |
ȼ ɨɛɳɟɦ ɫɥɭɱɚɟ ɭɪɚɜɧɟɧɢɟ ɫɜɹɡɢ ɦɨɠɧɨ ɩɪɟɞɫɬɚɜɢɬɶ ɜ ɜɢɞɟ |
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ra rN v v va vN t = |
ȿɫɥɢɭɪɚɜɧɟɧɢɟɫɜɹɡɢɧɟɫɨɞɟɪɠɢɬɫɤɨɪɨɫɬɟɣɱɚɫɬɢɰɫɢɫɬɟɦɵɬɨɫɜɹɡɶ ɧɚɡɵɜɚɟɬɫɹɝɨɥɨɧɨɦɧɨɣȼ ɩɪɨɬɢɜɧɨɦ ɫɥɭɱɚɟ ±ɧɟɝɨɥɨɧɨɦɧɨɣȿɫɥɢ ɭɪɚɜɧɟɧɢɟ ɫɜɹɡɢ ɧɟ ɫɨɞɟɪɠɢɬ ɹɜɧɨ ɜɪɟɦɹ ɬɨ ɫɜɹɡɶ ɧɚɡɵɜɚɟɬɫɹɫɬɚɰɢɨɧɚɪɧɨɣȼ ɩɪɨɬɢɜɧɨɦ ɫɥɭɱɚɟ ±ɧɟɫɬɚɰɢɨɧɚɪɧɨɣ
ɇɚɥɨɠɟɧɢɟ ɧɚ ɫɢɫɬɟɦɭ ɭɞɟɪɠɢɜɚɸɳɢɯ ɫɜɹɡɟɣ ɩɪɢɜɨɞɢɬ ɤ ɭɦɟɧɶɲɟɧɢɸ ɱɢɫɥɚ ɫɬɟɩɟɧɟɣ ɫɜɨɛɨɞɵ ɫɢɫɬɟɦɵɑɢɫɥɨ ɫɬɟɩɟɧɟɣ ɫɜɨɛɨɞɵ± ɷɬɨ ɱɢɫɥɨ ɧɟɡɚɜɢ ɫɢɦɵɯ ɩɚɪɚɦɟɬɪɨɜ ɨɞɧɨɡɧɚɱɧɨ ɨɩɪɟɞɟɥɹɸɳɢɯ ɩɨɥɨɠɟɧɢɟ ɫɢɫɬɟɦɵ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ ȼ ɤɚɱɟɫɬɜɟ ɬɚɤɢɯ ɩɚɪɚɦɟɬɪɨɜ ɜɵɫɬɭɩɚɸɬ ɤɨɨɪɞɢɧɚɬɚ ɭɝɨɥ ɢ ɞɪ ɉɨɥɨɠɟɧɢɟ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ ɫɜɨɛɨɞɧɨɣ ɱɚɫɬɢɰɵ ɨɩɪɟɞɟɥɹɟɬɫɹ ɬɪɟɦɹ ɩɚɪɚɦɟɬɪɚ ɦɢ ɩɪɨɫɬɪɚɧɫɬɜɨ ɬɪɟɯɦɟɪɧɨ ɫɜɨɛɨɞɧɨɣ ɫɢɫɬɟɦɵ ɢɡ N ɱɚɫɬɢɰ ± N ɩɚɪɚɦɟɬɪɚ ɦɢ ȿɫɥɢ ɧɚ ɫɢɫɬɟɦɭ ɱɚɫɬɢɰ ɧɚɥɨɠɟɧɨm ɭɞɟɪɠɢɜɚɸɳɢɯ ɫɜɹɡɟɣ ɬɨ ɱɢɫɥɨ ɧɟɡɚ ɜɢɫɢɦɵɯ ɩɚɪɚɦɟɬɪɨɜ ɨɩɪɟɞɟɥɹɸɳɢɯ ɩɨɥɨɠɟɧɢɟ ɫɢɫɬɟɦɵ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ ɛɭɞɟɬ
N − m
Ɉɛɳɚɹ ɡɚɞɚɱɚ ɞɢɧɚɦɢɤɢ ɞɥɹ ɫɢɫɬɟɦɵ ɱɚɫɬɢɰ ɫɨ ɫɜɹɡɹɦɢ
ɍɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɫɜɨɛɨɞɧɨɣ ɱɚɫɬɢɰɵ ɢɦɟɟɬ ɜɢɞ ma d r&a = F&a
dt
ɁɞɟɫɶF&a ɨɛɨɡɧɚɱɚɟɬ ɪɚɜɧɨɞɟɣɫɬɜɭɸɳɭɸ ɫɢɥ ɞɟɣɫɬɜɭɸɳɢɯ ɧɚ ɱɚɫɬɢɰɭ Ʉɨɝɞɚ ɞɜɢɠɟɧɢɟ ɱɚɫɬɢɰɵ ɨɝɪɚɧɢɱɟɧɨ ɫɜɹɡɶɸ ɤ ɫɢɥɟ F&a ɫɥɟɞɭɟɬ ɞɨɛɚɜɢɬɶ ɫɢɥɭ ɪɟɚɤɰɢɢ ɫɜɹɡɢ ɞɟɣɫɬɜɭɸɳɭɸ ɧɚ ɱɚɫɬɢɰɭ
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Лекция8.ОСНОВНЫЕПОНЯТИЯИТЕОРЕМЫАНАЛИТИЧЕСКОЙМЕХАНИКИ
ma d r&a = F&a + R&a dt
ɋɢɥɚF&a ɫɱɢɬɚɟɬɫɹ ɡɚɞɚɧɧɨɣ ɢ ɧɚɡɵɜɚɟɬɫɹ ɚɤɬɢɜɧɨɣ ɫɢɥɨɣ
Ɋɚɫɫɦɨɬɪɢɦɫɢɫɬɟɦɭɢɡ N ɱɚɫɬɢɰɧɚɤɨɬɨɪɭɸɧɚɤɥɚɞɵɜɚɸɬɫɹ m ɝɨɥɨ ɧɨɦɧɵɯ ɫɜɹɡɟɣ Ɍɨɝɞɚ ɡɚɞɚɱɚ ɨ ɧɚɯɨɠɞɟɧɢɢ ɞɜɢɠɟɧɢɹ ɱɚɫɬɢɰ ɫɢɫɬɟɦɵ ɫɜɨɞɢɬɫɹ
ɤ ɪɟɲɟɧɢɸ ɫɥɟɞɭɸɳɟɣ ɫɢɫɬɟɦɵ ɭɪɚɜɧɟɧɢɣ
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Ⱦɚɧɧɚɹ ɫɢɫɬɟɦɚ ɢɡ N +m ɭɪɚɜɧɟɧɢɣ ɫɨɞɟɪɠɢɬ N ɧɟɢɡɜɟɫɬɧɵɯ N ɤɨɨɪ ɞɢɧɚɬɱɚɫɬɢɰɢ N ɩɪɨɟɤɰɢɣɫɢɥɪɟɚɤɰɢɢɫɜɹɡɟɣȼ ɨɛɳɟɦɫɥɭɱɚɟ N +m < N ɢ ɡɚɞɚɱɚ ɧɟ ɹɜɥɹɟɬɫɹ ɨɩɪɟɞɟɥɟɧɧɨɣ Ⱦɚɧɧɚɹ ɡɚɞɚɱɚ ɩɨɥɧɨɫɬɶɸ ɨɩɪɟɞɟɥɟɧɚ ɜ ɬɨɦ ɫɥɭɱɚɟ ɤɨɝɞɚ ɫɜɹɡɢ ɹɜɥɹɸɬɫɹ ɢɞɟɚɥɶɧɵɦɢ
ȼɢɪɬɭɚɥɶɧɵɟ ɩɟɪɟɦɟɳɟɧɢɹ ɂɞɟɚɥɶɧɵɟ ɢ ɪɟɚɥɶɧɵɟ ɫɜɹɡɢ
Ƚɨɥɨɧɨɦɧɵɟ ɫɜɹɡɢ ɤɨɬɨɪɵɟ ɧɚɤɥɚɞɵɜɚɸɬɫɹ ɧɚ ɫɢɫɬɟɦɭ ɱɚɫɬɢɰ ɨɝɪɚɧɢɱɢ ɜɚɸɬ ɫɤɨɪɨɫɬɢ ɞɜɢɠɟɧɢɹ ɱɚɫɬɢɰ ɫɢɫɬɟɦɵ ɫɤɨɪɨɫɬɢ ɱɚɫɬɢɰ v&a ɞɨɥɠɧɵ ɭɞɨɜɥɟ
ɬɜɨɪɹɬɶ ɫɥɟɞɭɸɳɢɦ ɭɪɚɜɧɟɧɢɹɦ
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ɚ ɛɟɫɤɨɧɟɱɧɨ ɦɚɥɵɟ ɩɟɪɟɦɟɳɟɧɢɹ ɱɚɫɬɢɰdr&a |
= v&adt ɭɪɚɜɧɟɧɢɹɦ |
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∂fα |
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dt = |
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Ȼɟɫɤɨɧɟɱɧɨ ɦɚɥɵɟ ɩɟɪɟɦɟɳɟɧɢɹ dr&a ɹɜɥɹɸɬɫɹ ɜɨɡɦɨɠɧɵɦɢ ɩɟɪɟɦɟɳɟ
ɧɢɹɦɢ ɱɚɫɬɢɰ ɫɢɫɬɟɦɵ Ⱦɟɣɫɬɜɢɬɟɥɶɧɵɟ ɩɟɪɟɦɟɳɟɧɢɹ ɜɯɨɞɹɬ ɜ ɱɢɫɥɨ ɜɨɡɦɨɠ ɧɵɯ ɩɟɪɟɦɟɳɟɧɢɣ ȼɨɡɶɦɟɦ ɞɜɚ ɜɨɡɦɨɠɧɵɯ ɩɟɪɟɦɟɳɟɧɢɹ ɱɚɫɬɢɰɵ ɜɞɚɧɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢɢ ɩɪɟɞɫɬɚɜɢɦ ɢɯ ɪɚɡɧɨɫɬɶ
δ r&a = dr&a − dr&a
Ⱦɚɧɧɵɟ ɪɚɡɧɨɫɬɢ ɧɚɡɵɜɚɸɬɫɹɜɢɪɬɭɚɥɶɧɵɦɢ ɩɟɪɟɦɟɳɟɧɢɹɦɢɉɨɞɫɬɚɜɢɦ ɜɨɡɦɨɠɧɵɟ ɩɟɪɟɦɟɳɟɧɢɹ ɱɚɫɬɢɰɵdr&a dr&a ɜ ɭɪɚɜɧɟɧɢɹ ɢ ɜɵɱɬɟɦ ɫɨɨɬɜɟɬɫɬ
ɜɭɸɳɢɟ ɭɪɚɜɧɟɧɢɹ ɨɞɧɨ ɢɡ ɞɪɭɝɨɝɨ ȼ ɪɟɡɭɥɶɬɚɬɟ ɩɨɥɭɱɢɦ
N |
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ȼ ɦɚɬɟɦɚɬɢɤɟ ɜɟɥɢɱɢɧɵ ɩɨɞɨɛɧɵɟ ɢ ɥɟɜɨɣ ɱɚɫɬɢ ɧɚɡɵɜɚɸɬɜɚ ɪɢɚɰɢɹɦɢ ɩɨɷɬɨɦɭ δ r&a ɦɨɠɧɨ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɤɚɤ ɜɚɪɢɚɰɢɸ ɪɚɞɢɭɫ ɜɟɤɬɨɪɚ
ɱɚɫɬɢɰɵ ɚ ɥɟɜɭɸ ɱɚɫɬɶ ɤɚɤ ɜɚɪɢɚɰɢɸ ɮɭɧɤɰɢɢ ɋɜɹɡɶ ɧɚɡɵɜɚɟɬɫɹɢɞɟɚɥɶɧɨɣɟɫɥɢ ɪɚɛɨɬɚ ɫɢɥɵ ɪɟɚɤɰɢɢ ɫɜɹɡɢ ɧɚ ɥɸ
ɛɨɦ ɜɢɪɬɭɚɥɶɧɨɦ ɩɟɪɟɦɟɳɟɧɢɢ ɫɢɫɬɟɦɵ ɱɚɫɬɢɰɜɢɪɬɭɚɥɶɧɚɹ ɪɚɛɨɬɚɪɚɜɧɚ ɧɭɥɸ ɬ ɟ
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М. А. Михайлов. ЛЕКЦИИ ПО КЛАССИЧЕСКОЙ МЕХАНИКЕ
δ AR = ¦N R&aδ r&a =
a =
ɉɪɢɦɟɪɨɦ ɢɞɟɚɥɶɧɨɣ ɫɜɹɡɢ ɫɥɭɠɢɬ ɢɞɟɚɥɶɧɨ ɝɥɚɞɤɚɹ ɩɨɜɟɪɯɧɨɫɬɶ ɋɢɥɵ
ɪɟɚɤɰɢɢ ɫɜɹɡɢ ɜ ɞɚɧɧɨɦ ɫɥɭɱɚɟ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɵ ɩɨɜɟɪɯɧɨɫɬɢ ɢ ɫɥɟɞɨɜɚɬɟɥɶ
ɧɨ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɵ ɜɢɪɬɭɚɥɶɧɵɦ ɩɟɪɟɦɟɳɟɧɢɹɦ ɱɚɫɬɢɰ ɫɢɫɬɟɦɵ ɧɚ ɞɚɧɧɨɣ
ɩɨɜɟɪɯɧɨɫɬɢ ɌɟɨɪɟɦɚɈɫɧɨɜɧɚɹ ɡɚɞɚɱɚ ɞɢɧɚɦɢɤɢ ɫɢɫɬɟɦɵ ɱɚɫɬɢɰ ɫɨ ɫɜɹɡɹɦɢ ɩɨɥɧɨ
ɫɬɶɸ ɨɩɪɟɞɟɥɟɧɚ ɟɫɥɢ ɫɜɹɡɢ ɹɜɥɹɸɬɫɹ ɢɞɟɚɥɶɧɵɦɢ Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨɍɦɧɨɠɢɦ ɤɚɠɞɨɟ ɢɡ ɭɪɚɜɧɟɧɢɣ ɧɚ ɧɟɨɩɪɟɞɟɥɟɧɧɵɟ
ɤɨɷɮɮɢɰɢɟɧɬɵ ±λα ɫɥɨɠɢɦ ɩɨɥɭɱɟɧɧɵɟ ɭɪɚɜɧɟɧɢɹ ɢ ɭɱɬɟɦ ɭɫɥɨɜɢɟ ɢɞɟɚɥɶɧɨ ɫɬɢ ɫɜɹɡɟɣ ȼ ɪɟɡɭɥɶɬɚɬɟ ɩɨɥɭɱɚɟɦ
N § |
& |
m |
∂fα |
· |
& |
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¦¨¨Ra − ¦λα |
& |
¸¸ |
δ ra |
= |
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a= © |
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α = |
∂ra |
¹ |
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ɂɡ N ɤɨɦɩɨɧɟɧɬ ɜɢɪɬɭɚɥɶɧɵɯ ɩɟɪɟɦɟɳɟɧɢɣ ɜ ɞɚɧɧɨɦ ɭɪɚɜɧɟɧɢɢ m ɹɜ ɥɹɟɬɫɹ ɡɚɜɢɫɢɦɵɦ ɢ N − m ɹɜɥɹɸɬɫɹ ɧɟɡɚɜɢɫɢɦɵɦɢ Ⱦɚɥɟɟ ɤɨɷɮɮɢɰɢɟɧɬɵ λα
ɩɨɞɛɟɪɟɦ ɬɚɤ ɱɬɨɛɵ ɜɵɪɚɠɟɧɢɹ ɜ ɫɤɨɛɤɚɯ ɩɟɪɟɞ ɡɚɜɢɫɢɦɵɦɢ ɜɚɪɢɚɰɢɹɦɢ ɨɛɪɚɬɢɥɢɫɶ ɜ ɧɭɥɶ ɉɨɫɥɟ ɬɚɤɨɣ ɩɪɨɰɟɞɭɪɵ ɜ ɫɭɦɦɟ ɨɫɬɚɧɭɬɫɹ N − m ɫɥɚ ɝɚɟɦɵɯ ɤɨɬɨɪɵɟ ɫɨɞɟɪɠɚɬ ɬɨɥɶɤɨ ɜɚɪɢɚɰɢɢ ɧɟɡɚɜɢɫɢɦɵɯ ɤɨɨɪɞɢɧɚɬ ɉɨɫɤɨɥɶɤɭ ɡɧɚɱɟɧɢɹ ɜɢɪɬɭɚɥɶɧɵɯ ɩɟɪɟɦɟɳɟɧɢɣ ɧɟɡɚɜɢɫɢɦɵɯ ɤɨɦɩɨɧɟɧɬ ɹɜɥɹɸɬɫɹ ɩɪɨɢɡ ɜɨɥɶɧɵɦɢ ɬɨ ɤɨɷɮɮɢɰɢɟɧɬɵ ɜɵɪɚɠɟɧɢɹ ɜ ɫɤɨɛɤɚɯ ɩɪɢ ɧɢɯ ɞɨɥɠɧɵ ɨɛɪɚ ɳɚɬɶɫɹ ɜ ɧɭɥɶ ȼ ɪɟɡɭɥɶɬɚɬɟ ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ
m |
∂f&α |
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R&a = ¦λα |
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α = |
∂ra |
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Ⱦɚɥɟɟ ɧɭɠɧɨɩɨɞɫɬɚɜɢɬɶɜ ɫɢɫɬɟɦɭɭɪɚɜɧɟɧɢɣ ȼ ɪɟɡɭɥɶɬɚɬɟ ɩɨɥɭɱɢɦ ɫɢɫɬɟɦɭ ɭɪɚɜɧɟɧɢɣ
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d r&a |
& m |
∂fα |
& & |
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& |
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ma |
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= Fa + ¦λα |
& |
a = N fα r r |
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rN t = |
α = m |
dt |
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α = |
∂ra |
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Ⱦɚɧɧɚɹ ɫɢɫɬɟɦɚ ɭɪɚɜɧɟɧɢɣ ɩɨɥɧɨɫɬɶɸ ɨɩɪɟɞɟɥɟɧɚ ɬɚɤ ɤɚɤ ɨɧɚ ɫɨɞɟɪɠɢɬN + m ɭɪɚɜɧɟɧɢɣɨɬɧɨɫɢɬɟɥɶɧɨɬɚɤɨɝɨɠɟɱɢɫɥɚɧɟɢɡɜɟɫɬɧɵɯ N ɤɨɨɪɞɢɧɚɬ ɢm ɦɧɨɠɢɬɟɥɟɣλα
ɉɪɢɧɰɢɩ ɜɢɪɬɭɚɥɶɧɵɯ ɩɟɪɟɦɟɳɟɧɢɣ
ȼ ɩɨɥɨɠɟɧɢɢ ɪɚɜɧɨɜɟɫɢɹ ɪɚɛɨɬɚ ɚɤɬɢɜɧɵɯ ɫɢɥ ɧɚ ɥɸɛɵɯ ɜɢɪɬɭɚɥɶɧɵɯ ɩɟ
ɪɟɦɟɳɟɧɢɹɯ ɱɚɫɬɢɰ ɫɢɫɬɟɦɵ ɪɚɜɧɚ ɧɭɥɸ
δ AA = ¦N F&aδ r&a =
a =
ɍɪɚɜɧɟɧɢɟ ɦɚɬɟɦɚɬɢɱɟɫɤɢɜɵɪɚɠɚɟɬɫɨɛɨɣɩɪɢɧɰɢɩɜɢɪɬɭɚɥɶɧɵɯ ɩɟɪɟɦɟɳɟɧɢɣ ɍɫɥɨɜɢɟ ɧɟɨɛɯɨɞɢɦɨ ɢ ɞɨɫɬɚɬɨɱɧɨ ɞɥɹ ɬɨɝɨ ɱɬɨɛɵ ɫɢɫɬɟɦɚ
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Лекция8.ОСНОВНЫЕПОНЯТИЯИТЕОРЕМЫАНАЛИТИЧЕСКОЙМЕХАНИКИ
ɱɚɫɬɢɰ ɫ ɝɨɥɨɧɨɦɧɵɦɢ ɢɞɟɚɥɶɧɵɦɢ ɭɞɟɪɠɢɜɚɸɳɢɦɢ ɢ ɫɬɚɰɢɨɧɚɪɧɵɦɢ ɫɜɹɡɹ
ɦɢ ɧɚɯɨɞɢɥɚɫɶ ɜ ɫɨɫɬɨɹɧɢɢ ɪɚɜɧɨɜɟɫɢɹ
Ⱦɢɮɮɟɪɟɧɰɢɚɥɶɧɵɣ ɜɚɪɢɚɰɢɨɧɧɵɣ ɩɪɢɧɰɢɩ Ⱦ¶Ⱥɥɚɦɛɟɪɚ ± Ʌɚɝɪɚɧɠɚ
Ɋɚɫɫɦɨɬɪɢɦɫɢɫɬɟɦɭɢɡ N ɱɚɫɬɢɰɧɚɤɨɬɨɪɭɸɧɚɤɥɚɞɵɜɚɸɬɫɹ m ɝɨɥɨ ɧɨɦɧɵɯ ɢɞɟɚɥɶɧɵɯ ɫɜɹɡɟɣ Ⱦɥɹ ɬɚɤɨɣ ɫɢɫɬɟɦɵ ɱɚɫɬɢɰ ɫɩɪɚɜɟɞɥɢɜɚ ɫɢɫɬɟɦɚ ɭɪɚɜɧɟɧɢɣ ɍɦɧɨɠɢɦ ɤɚɠɞɨɟ ɢɡ ɭɪɚɜɧɟɧɢɣ ɞɜɢɠɟɧɢɹ ɜ ɫɤɚɥɹɪɧɨ ɧɚ ɜɢɪɬɭɚɥɶɧɨɟ ɩɟɪɟɦɟɳɟɧɢɟδ r&a ɫɥɨɠɢɦ ɩɨɥɭɱɟɧɧɵɟ ɭɪɚɜɧɟɧɢɹ ɢ ɭɱɬɟɦ ɭɫɥɨɜɢɟ
ɢɞɟɚɥɶɧɨɫɬɢ ɫɜɹɡɟɣ ȼ ɪɟɡɭɥɶɬɚɬɟ ɩɨɥɭɱɢɦ
N § |
& |
& · |
& |
= |
¦¨Fa |
− ma va ¸ |
δ ra |
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a − © |
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¹ |
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Ɂɞɟɫɶ ɢ ɜ ɞɚɥɶɧɟɣɲɟɦ ɬɨɱɤɨɣ ɧɚɞ ɩɟɪɟɦɟɧɧɨɣ ɛɭɞɟɦ ɨɛɨɡɧɚɱɚɬɶ ɩɨɥɧɭɸ
ɩɪɨɢɡɜɨɞɧɭɸ ɩɨ ɜɪɟɦɟɧɢ ɍɪɚɜɧɟɧɢɟ ɹɜɥɹɟɬɫɹ ɦɚɬɟɦɚɬɢɱɟɫɤɢɦ ɜɵɪɚɠɟɧɢɟɦ ɜɚɪɢɚɰɢɨɧɧɨɝɨ
ɩɪɢɧɰɢɩɚ Ⱦ¶Ⱥɥɚɦɛɟɪɚ ± Ʌɚɝɪɚɧɠɚɟɫɥɢ ɧɚ ɦɟɯɚɧɢɱɟɫɤɭɸ ɫɢɫɬɟɦɭ ɧɚɤɥɚɞɵɜɚ ɸɬɫɹ ɝɨɥɨɧɨɦɧɵɟ ɢɞɟɚɥɶɧɵɟ ɫɜɹɡɢ ɬɨ ɜ ɤɚɠɞɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ɞɨɥɠɧɨ ɜɵ ɩɨɥɧɹɬɶɫɹ ɭɫɥɨɜɢɟ
ɉɪɢɦɟɪ ɋɢɫɬɟɦɚɬɟɥɩɨɤɚɡɚɧɧɚɹɧɚɪɢɫ ɫɜɹɡɚɧɚɧɢɬɶɸɩɟɪɟ ɛɪɨɲɟɧɧɨɣ ɱɟɪɟɡ ɛɥɨɤ ɉɪɟɧɟɛɪɟɝɚɹ ɢɧɟɪɬɧɵɦɢ ɫɜɨɣɫɬɜɚɦɢ ɛɥɨɤɚ ɧɚɣɬɢ ɭɫɤɨ ɪɟɧɢɹ ɬɟɥ
δ r& m Γ U& P
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δ r& |
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Γ U& |
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δ r& |
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Γ U |
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m g |
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P J& |
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P
Ɋɢɫ. 8.1.Ɉɞɧɨ ɢɡ ɜɨɡɦɨɠɧɵɯ ɩɟɪɟɦɟɳɟɧɢɣ ɝɪɭɡɨɜ ɧɚ ɧɟɩɨɞɜɢɠɧɨɦ ɛɥɨɤɟ
49
М. А. Михайлов. ЛЕКЦИИ ПО КЛАССИЧЕСКОЙ МЕХАНИКЕ
Ɂɚɩɢɲɟɦ ɭɪɚɜɧɟɧɢɟ ɞɥɹ ɞɚɧɧɨɣ ɫɢɫɬɟɦɵ ɬɟɥ
− m gδr − m gδr + m gδr − m |
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d r |
δr − m |
d r |
δr − m |
d r |
δr = |
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dt |
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ɋ ɭɱɟɬɨɦ ɫɜɹɡɟɣ |
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δ r = δ r = δ r |
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= w |
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ɧɚɯɨɞɢɦ ɭɫɤɨɪɟɧɢɟ
w = g(m − m − m ) m + m + m
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