Лекции по классической механике. Учебное пособие
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ɅȿɄɐɂə ɈɋɇɈȼɕ ɆȿɏȺɇɂɄɂ ɅȺȽɊȺɇɀȺ
Ɉɛɨɛɳɟɧɧɵɟ ɤɨɨɪɞɢɧɚɬɵ ɢ ɨɛɨɛɳɟɧɧɵɟ ɫɢɥɵ
ɉɚɪɚɦɟɬɪɵ ɤɨɬɨɪɵɟ ɨɩɪɟɞɟɥɹɸɬ ɩɨɥɨɠɟɧɢɟ ɦɟɯɚɧɢɱɟɫɤɨɣ ɫɢɫɬɟɦɵ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ ɧɚɡɵɜɚɸɬɫɹ ɨɛɨɛɳɟɧɧɵɦɢ ɤɨɨɪɞɢɧɚɬɚɦɢ ɢ ɨɛɨɡɧɚɱɚɸɬɫɹ q q qk qs ɝɞɟ ɢɧɞɟɤɫ k ɩɪɢɧɢɦɚɟɬ ɡɧɚɱɟɧɢɹ ɨɬ ɞɨ s ± ɱɢɫɥɨ ɫɬɟɩɟɧɟɣ
ɫɜɨɛɨɞɵ
Ⱦɟɤɚɪɬɨɜɵ ɤɨɨɪɞɢɧɚɬɵ ɱɚɫɬɢɰ ɦɨɝɭɬ ɛɵɬɶ ɜɵɪɚɠɟɧɵ ɱɟɪɟɡ ɨɛɨɛɳɟɧɧɵɟ
ɤɨɨɪɞɢɧɚɬɵ
r&a = r&a q qs t a = N
ɉɪɢɦɟɪɋɮɟɪɢɱɟɫɤɢɣ ɦɚɹɬɧɢɤ ɞɥɢɧɵ l Ⱦɟɤɚɪɬɨɜɵ ɤɨɨɪɞɢɧɚɬɵ ɜɵ ɪɚɠɚɸɬɫɹ ɱɟɪɟɡ ɫɮɟɪɢɱɟɫɤɢɟ ɤɨɨɪɞɢɧɚɬɵ ϕ θ ɤɨɬɨɪɵɟ ɦɨɝɭɬ ɜɵɫɬɭɩɚɬɶ ɜ ɤɚɱɟɫɬɜɟ ɨɛɨɛɳɟɧɧɵɯ ɤɨɨɪɞɢɧɚɬ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ
x = l VLQθ FRVϕ y = l VLQθ VLQϕ z = l FRVθ
ɭɪɚɜɧɟɧɢɟ ɫɜɹɡɢ ɢɦɟɟɬ ɜɢɞ ɜ ɞɟɤɚɪɬɨɜɵɯ ɤɨɨɪɞɢɧɚɬɚɯ ±x + y + z = l ɜ ɫɮɟɪɢɱɟɫɤɢɯ ɤɨɨɪɞɢɧɚɬɚɯ ±r = l
Ⱦɚɧɧɚɹ ɫɢɫɬɟɦɚ ɢɦɟɟɬ ɞɜɟ ɫɬɟɩɟɧɢ ɫɜɨɛɨɞɵ ɞɜɢɠɟɧɢɹ
ȼɢɪɬɭɚɥɶɧɵɟ ɩɟɪɟɦɟɳɟɧɢɹ ɨɩɪɟɞɟɥɹɸɬɫɹ ɫɥɟɞɭɸɳɢɦɢ ɫɨɨɬɧɨɲɟɧɢɹɦɢ
δ r&a = ¦s ∂r&a δ qk
k = ∂qk
ȼɵɪɚɡɢɦ ɜɢɪɬɭɚɥɶɧɭɸ ɪɚɛɨɬɭ ɚɤɬɢɜɧɵɯ ɫɢɥ ɱɟɪɟɡ ɨɛɨɛɳɟɧɧɵɟ ɤɨɨɪɞɢɧɚ
ɬɵ ɱɚɫɬɢɰ
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ɟɫɬɶ ɨɛɨɛɳɟɧɧɵɟ ɫɢɥɵ
51
М. А. Михайлов. ЛЕКЦИИ ПО КЛАССИЧЕСКОЙ МЕХАНИКЕ
ɍɪɚɜɧɟɧɢɹ Ʌɚɝɪɚɧɠɚ ɝɨ ɪɨɞɚ
ɉɨɥɭɱɢɦ ɭɪɚɜɧɟɧɢɹ Ʌɚɝɪɚɧɠɚ ɝɨ ɪɨɞɚ ɢɡ ɩɪɢɧɰɢɩɚ Ⱦ¶Ⱥɥɚɦɛɟɪɚ ± Ʌɚ
ɝɪɚɧɠɚ Ⱦɥɹ ɷɬɨɝɨ ɩɨɞɫɬɚɜɢɦ ɜ ɜɵɪɚɠɟɧɢɹ ɢ ɢɡɦɟɧɢɦ ɩɨɪɹɞɨɤ ɫɭɦɦɢɪɨɜɚɧɢɹ ɩɨ ɢɧɞɟɤɫɚɦa k ȼ ɪɟɡɭɥɶɬɚɬɟ ɩɨɥɭɱɢɦ
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ȼɜɟɞɟɦ ɨɛɨɡɧɚɱɟɧɢɹ |
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ɋ ɭɱɟɬɨɦ ɨɩɪɟɞɟɥɟɧɢɹ ɢ ɨɛɨɡɧɚɱɟɧɢɣ ɮɨɪɦɭɥɭ ɩɟɪɟɩɢɲɟɦ ɜ ɜɢɞɟ
s
¦(Qk − Ak )δ qk =
k =
Ⱦɥɹ ɜɵɩɨɥɧɟɧɢɹ ɪɚɜɟɧɫɬɜɚ ɜ ɫɢɥɭ ɧɟɡɚɜɢɫɢɦɨɫɬɢ ɜɚɪɢɚɰɢɣ ɨɛɨɛ
ɳɟɧɧɵɯ ɤɨɨɪɞɢɧɚɬ ɞɨɥɠɧɨ ɛɵɬɶ
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ɉɪɟɨɛɪɚɡɭɟɦ ɪɚɜɟɧɫɬɜɨ ɢɫɩɨɥɶɡɭɹ ɬɨɠɞɟɫɬɜɚ |
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ɢ ɪɚɜɟɧɫɬɜɨ |
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ȼ ɪɟɡɭɥɶɬɚɬɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ ɩɨɥɭɱɚɟɦ |
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ɤɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɫɢɫɬɟɦɵ ɱɚɫɬɢɰ
ɉɨɞɫɬɚɜɢɜ ɮɨɪɦɭɥɭ ɜ ɭɪɚɜɧɟɧɢɟ ɩɨɥɭɱɢɦ ɢɫɤɨɦɵɟ ɭɪɚɜɧɟɧɢɹ
Ʌɚɝɪɚɧɠɚ ɝɨ ɪɨɞɚ
Qk = |
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Ɋɚɫɫɦɨɬɪɢɦɫɥɭɱɚɣɩɨɬɟɧɰɢɚɥɶɧɵɯɫɢɥ F&a = − dU
dr&a ȼ ɷɬɨɦɫɥɭɱɚɟɞɥɹ ɨɛɨɛɳɟɧɧɵɯ ɫɢɥ ɩɨɥɭɱɚɟɦ
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Лекция9.ОСНОВЫМЕХАНИКИЛАГРАНЖА |
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ɉɨɞɫɬɚɜɢɜ ɜ ɭɪɚɜɧɟɧɢɹ Ʌɚɝɪɚɧɠɚ ɩɨɥɭɱɢɦ |
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Ɂɞɟɫɶ ɮɭɧɤɰɢɹ
L = T qk qk t − U q t
ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɪɚɡɧɨɫɬɶ ɤɢɧɟɬɢɱɟɫɤɨɣ ɢ ɩɨɬɟɧɰɢɚɥɶɧɨɣ ɷɧɟɪɝɢɢ ɫɢɫɬɟɦɵ
ɱɚɫɬɢɰ ɢ ɜɵɪɚɠɚɟɬɫɹ ɱɟɪɟɡ ɨɛɨɛɳɟɧɧɵɟ ɤɨɨɪɞɢɧɚɬɵ ɫɤɨɪɨɫɬɢ ɱɚɫɬɢɰ ɫɢɫɬɟɦɵ ɢ ɜɪɟɦɹ Ɏɭɧɤɰɢɹ ɧɚɡɵɜɚɟɬɫɹɮɭɧɤɰɢɟɣ Ʌɚɝɪɚɧɠɚ
Ɏɭɧɤɰɢɹ Ʌɚɝɪɚɧɠɚ ɡɚɞɚɟɬɫɹ ɧɟɨɞɧɨɡɧɚɱɧɨ ȼ ɱɚɫɬɧɨɫɬɢ ɢɡ ɭɪɚɜɧɟɧɢɣ ɫɥɟɞɭɟɬ ɱɬɨ ɩɪɢɛɚɜɥɟɧɢɟ ɤ ɧɟɣ ɥɸɛɨɣ ɜɟɥɢɱɢɧɵ ɤɨɬɨɪɚɹ ɧɟ ɡɚɜɢɫɢɬ ɹɜɧɨ ɨɬ ɨɛɨɛɳɟɧɧɵɯ ɤɨɨɪɞɢɧɚɬ ɢ ɫɤɨɪɨɫɬɟɣ ɱɚɫɬɢɰ ɧɟ ɦɟɧɹɟɬ ɭɪɚɜɧɟɧɢɣ ɋɥɟɞɭɟɬ ɡɚɦɟɬɢɬɶ ɱɬɨ ɜɢɞ ɭɪɚɜɧɟɧɢɣ Ʌɚɝɪɚɧɠɚ ɧɟ ɡɚɜɢɫɢɬ ɨɬ ɜɵɛɨɪɚ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ ɢ ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ ɬ ɟ ɞɚɧɧɵɟ ɭɪɚɜɧɟɧɢɹ ɢɧɜɚɪɢɚɧɬɧɵ ɩɨ ɨɬɧɨɲɟ ɧɢɸ ɤ ɜɵɛɨɪɭ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ ɢ ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ
ɉɪɢɦɟɪ ɑɚɫɬɢɰɚ ɜ ɩɨɥɟ ɫɢɥɵ ɬɹɠɟɫɬɢ Ɂɟɦɥɢ
Ⱦɥɹ ɞɚɧɧɨɣ ɱɚɫɬɢɰɵ ɤɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹT = m v ɩɨɬɟɧɰɢɚɥɶɧɚɹ ɷɧɟɪ
ɝɢɹ ɪɚɜɧɚmgh Ɏɭɧɤɰɢɹ Ʌɚɝɪɚɧɠɚ ɢɦɟɟɬ ɜɢɞ
L = T = m vx + vy + vz − mgz
ȼ ɤɚɱɟɫɬɜɟ ɨɛɨɛɳɟɧɧɵɯ ɤɨɨɪɞɢɧɚɬ ɡɞɟɫɶ ɜɵɛɪɚɧɵ ɞɟɤɚɪɬɨɜɵ ɤɨɨɪɞɢɧɚ
ɬɵ ɉɨɞɫɬɚɜɥɹɹ ɮɭɧɤɰɢɸ Ʌɚɝɪɚɧɠɚ ɜ ɭɪɚɜɧɟɧɢɹ ɧɚɣɞɟɦ
d |
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ɗɬɨ ɟɫɬɶ ɭɪɚɜɧɟɧɢɹ ɞɜɢɠɟɧɢɹ ɱɚɫɬɢɰɵ ɜ ɩɨɥɟ ɫɢɥɵ ɬɹɠɟɫɬɢ ɂɧɬɟɝɪɢɪɭɹ
ɞɚɧɧɵɟ ɭɪɚɜɧɟɧɢɹ ɧɚɯɨɞɢɦ ɡɚɤɨɧ ɞɜɢɠɟɧɢɹ ɱɚɫɬɢɰɵɉɪɭɠɢɧɧɵɣ ɦɚɹɬɧɢɤȾɥɹ ɱɚɫɬɢɰɵ ɦɚɫɫɵm ɫɨɜɟɪɲɚɸɳɟɣ ɤɨɥɟɛɚɧɢɹ
ɧɚ ɩɪɭɠɢɧɟ ɠɟɫɬɤɨɫɬɶɸk ɮɭɧɤɰɢɹ Ʌɚɝɪɚɧɠɚ ɢɦɟɟɬ ɜɢɞ
L= m vx − k x
ɝɞɟ x ± ɫɦɟɳɟɧɢɟ ɱɚɫɬɢɰɵ ɢɡ ɩɨɥɨɠɟɧɢɹ ɪɚɜɧɨɜɟɫɢɹ ɉɨɞɫɬɚɜɢɜ ɞɚɧɧɭɸ ɮɭɧɤ ɰɢɸ ɜ ɩɨɥɭɱɢɦ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ ɝɚɪɦɨɧɢɱɟɫɤɢɯ ɤɨɥɟɛɚɧɢɣ
ddt mvx + kx =
53
М. А. Михайлов. ЛЕКЦИИ ПО КЛАССИЧЕСКОЙ МЕХАНИКЕ
Ʉɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɟɟ ɜɵɪɚɠɟɧɢɟ ɱɟɪɟɡ ɨɛɨɛɳɟɧɧɵɟ ɤɨɨɪɞɢɧɚɬɵ
ɢ ɫɤɨɪɨɫɬɢ
Ɋɚɫɫɦɨɬɪɢɦ ɦɟɯɚɧɢɱɟɫɤɭɸ ɫɢɫɬɟɦɭ ɧɚ ɤɨɬɨɪɭɸ ɧɚɤɥɚɞɵɜɚɸɬɫɹ ɧɟɫɬɚ ɰɢɨɧɚɪɧɵɟ ɫɜɹɡɢ ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɫɤɨɪɨɫɬɢ ɨɬɞɟɥɶɧɵɯ ɱɚɫɬɢɰ ɫɢɫɬɟɦɵ ɩɪɟɞɫɬɚɜ
ɥɹɸɬɫɹ ɱɟɪɟɡ ɨɛɨɛɳɟɧɧɵɟ ɫɤɨɪɨɫɬɢ ɩɨ ɮɨɪɦɭɥɚɦ |
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ɉɨɞɫɬɚɜɢɦ ɞɚɧɧɵɟ ɮɨɪɦɭɥɵ ɜ ȼ ɪɟɡɭɥɶɬɚɬɟ ɩɨɥɭɱɢɦ |
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ɂɡɦɟɧɢɦ ɩɨɪɹɞɨɤ ɫɭɦɦɢɪɨɜɚɧɢɹ ɜ ɩɨɫɥɟɞɧɟɣ ɮɨɪɦɭɥɟ Ɉɤɨɧɱɚɬɟɥɶɧɨ ɩɨɥɭɱɢɦ |
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ɝɞɟ |
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Ɂɞɟɫɶ ɮɭɧɤɰɢɢ f ɢT ɹɜɧɨ ɡɚɜɢɫɹɬ ɨɬ ɨɛɨɛɳɟɧɧɵɯ ɤɨɨɪɞɢɧɚɬ ɱɚɫɬɢɰ ɢ ɜɪɟɦɟɧɢ
ȼ ɪɟɡɭɥɶɬɚɬɟ ɦɨɠɧɨ ɫɞɟɥɚɬɶ ɫɥɟɞɭɸɳɢɟ ɜɵɜɨɞɵ ɤɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɫɢɫɬɟɦɵ ɱɚɫɬɢɰ ɹɜɥɹɟɬɫɹ ɧɟɨɞɧɨɪɨɞɧɨɣ ɤɜɚɞɪɚɬɢɱɧɨɣ ɮɨɪɦɨɣ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɛɨɛɳɟɧɧɵɯ ɫɤɨɪɨɫɬɟɣ ɤɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɢ ɮɭɧɤɰɢɹ Ʌɚɝɪɚɧɠɚ ɹɜɧɨ ɡɚ ɜɢɫɹɬ ɨɬ ɜɪɟɦɟɧɢ
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 ɧɚɡɜɚɧɢɟ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɢɧɟɪɰɢɢ ȼ ɤɚɱɟɫɬɜɟ ɢɯ ɦɨɝɭɬ ɜɵɫɬɭɩɚɬɶ ɧɚɩɪɢɦɟɪ ɦɚɫɫɵ ɢ ɦɨɦɟɧɬɵ ɢɧɟɪɰɢɢ ɱɚɫɬɢɰ ɫɢɫɬɟɦɵ
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Лекция9.ОСНОВЫМЕХАНИКИЛАГРАНЖА
ɉɪɢɦɟɪɊɚɫɫɦɨɬɪɢɦ ɫɢɫɬɟɦɭ ɢɡ ɞɜɭɯ ɱɚɫɬɢɰ ȼ ɤɚɱɟɫɬɜɟ ɨɛɨɛɳɟɧɧɵɯ ɤɨɨɪɞɢɧɚɬ ɜɵɛɟɪɟɦ ɪɚɞɢɭɫ ɜɟɤɬɨɪ ɰɟɧɬɪɚ ɢɧɟɪɰɢɢ ɫɢɫɬɟɦɵR&c ɢ ɜɟɤɬɨɪr& = r& − r& Ʉɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɞɚɧɧɨɣ ɫɢɫɬɟɦɵ ɢɦɟɟɬ ɜɢɞ
T= m R&c + μ r&
ɝɞɟm ± ɦɚɫɫɚ ɫɢɫɬɟɦɵμ ± ɩɪɢɜɟɞɟɧɧɚɹ ɦɚɫɫɚ ɫɢɫɬɟɦɵ
ɉɪɢɧɰɢɩ ɷɤɫɬɪɟɦɚɥɶɧɨɝɨ ɞɟɣɫɬɜɢɹ
ɉɭɫɬɶ ɩɨɥɨɠɟɧɢɟ ɦɟɯɚɧɢɱɟɫɤɨɣ ɫɢɫɬɟɦɵ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ ɡɚɞɚɟɬɫɹ ɫ ɩɨɦɨɳɶɸ ɨɛɨɛɳɟɧɧɵɯ ɤɨɨɪɞɢɧɚɬ ± qk Ʉɨɧɮɢɝɭɪɚɰɢɨɧɧɵɦ ɩɪɨɫɬɪɚɧɫɬɜɨɦ
ɦɟɯɚɧɢɱɟɫɤɨɣɫɢɫɬɟɦɵɧɚɡɵɜɚɟɬɫɹ s ɦɟɪɧɨɟɩɪɨɫɬɪɚɧɫɬɜɨɬɨɱɤɚɦɢɤɨɬɨɪɨɝɨ ɹɜɥɹɸɬɫɹ q q qk qs ɋɨɫɬɨɹɧɢɸ ɫɢɫɬɟɦɵ ɜ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ t ɨɬɜɟɱɚɟɬ
ɬɨɱɤɚ ɜ ɤɨɧɮɢɝɭɪɚɰɢɨɧɧɨɦ ɩɪɨɫɬɪɚɧɫɬɜɟ ɉɟɪɟɯɨɞ ɫɢɫɬɟɦɵ ɢɡ ɫɨɫɬɨɹɧɢɹ A ɜ ɫɨɫɬɨɹɧɢɟ B ɦɨɠɧɨ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɤɚɤ ɞɜɢɠɟɧɢɟ ɢɡɨɛɪɚɠɚɸɳɟɣ ɬɨɱɤɢ ɜ ɤɨɧɮɢɝɭɪɚɰɢɨɧɧɨɦ ɩɪɨɫɬɪɚɧɫɬɜɟ
ȼɜɟɞɟɦɮɭɧɤɰɢɸ ɞɟɣɫɬɜɢɹ S ɩɨ ɫɥɟɞɭɸɳɟɦɭ ɨɩɪɟɞɟɥɟɧɢɸ
t
S = ³L qk qk t dt
t
Ɏɭɧɤɰɢɹ ɞɟɣɫɬɜɢɹ ɢɦɟɟɬ ɪɚɡɦɟɪɧɨɫɬɶ >ɷɧɟɪɝɢɹ× ɜɪɟɦɹ@ ɤɨɬɨɪɚɹ ɫɨɜɩɚɞɚɟɬ ɫ ɪɚɡɦɟɪɧɨɫɬɶɸ ɦɨɦɟɧɬɚ ɢɦɩɭɥɶɫɚ
ɉɪɢɧɰɢɩ ɷɤɫɬɪɟɦɚɥɶɧɨɝɨ ɞɟɣɫɬɜɢɹ ɩɪɢɧɰɢɩ Ƚɚɦɢɥɶɬɨɧɚ ɭɬɜɟɪɠɞɚɟɬ ɱɬɨ ɩɪɢ ɩɟɪɟɯɨɞɟ ɫɢɫɬɟɦɵ ɢɡ ɫɨɫɬɨɹɧɢɹ A ɜ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢt ɜ ɫɨɫɬɨɹɧɢɟB ɜ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢt ɪɟɚɥɶɧɨ ɨɫɭɳɟɫɬɜɥɹɟɬɫɹ ɬɚɤɨɣ ɩɭɬɶ ɢɡɨɛɪɚɠɚɸɳɟɣ ɬɨɱɤɢ ɜ ɤɨɧɮɢɝɭɪɚɰɢɨɧɧɨɦ ɩɪɨɫɬɪɚɧɫɬɜɟ ɩɪɢ ɤɨɬɨɪɨɦ ɮɭɧɤɰɢɹ ɞɟɣɫɬɜɢɹ ɢɦɟɟɬ ɷɤɫ ɬɪɟɦɚɥɶɧɨɟ ɡɧɚɱɟɧɢɟ ɤɚɤ ɩɪɚɜɢɥɨ ɦɢɧɢɦɚɥɶɧɨɟ Ⱦɪɭɝɢɦɢ ɫɥɨɜɚɦɢ ɜɚɪɢɚɰɢɹ ɮɭɧɤɰɢɢ ɞɟɣɫɬɜɢɹ ɨɛɪɚɳɚɟɬɫɹ ɜ ɧɭɥɶ
δ S =
Ⱦɚɧɧɵɣ ɩɪɢɧɰɢɩ ɫɩɪɚɜɟɞɥɢɜ ɜ ɫɥɭɱɚɟ ɦɟɯɚɧɢɱɟɫɤɨɣ ɫɢɫɬɟɦɵ ɫ ɝɨɥɨɧɨɦɧɵɦɢ ɢɞɟɚɥɶɧɵɦɢ ɫɜɹɡɹɦɢ ɢ ɚɤɬɢɜɧɵɦɢ ɩɨɬɟɧɰɢɚɥɶɧɵɦɢ ɫɢɥɚɦɢ Ɂɚɦɟɬɢɦ ɱɬɨ ɞɚɧɧɵɣ ɩɪɢɧɰɢɩ ɫɩɪɚɜɟɞɥɢɜ ɢ ɜ ɫɥɭɱɚɟ ɨɛɨɛɳɟɧɧɨ
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ɩɨɬɟɧɰɢɚɥɶɧɵɯ ɫɢɥ ɤɨɬɨɪɵɟ ɩɪɟɞɫɬɚɜɥɹɸɬɫɹ ɜ ɜɢɞɟ |
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V = V qk qk t ± ɨɛɨɛɳɟɧɧɵɣ ɩɨɬɟɧɰɢɚɥ |
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55
М. А. Михайлов. ЛЕКЦИИ ПО КЛАССИЧЕСКОЙ МЕХАНИКЕ
Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨɇɚ ɫɚɦɨɦ ɞɟɥɟ
t |
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δ S = δ ³Ldt = ³δ Ldt = ³¦¨ |
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Ɂɞɟɫɶ ɢɫɩɨɥɶɡɨɜɚɧɨ ɨɩɪɟɞɟɥɟɧɢɟ ɜɚɪɢɚɰɢɢ ɮɭɧɤɰɢɢ ɂɧɬɟɝɪɢɪɭɟɦ ɩɨ ɱɚɫ
ɬɹɦ ɞɚɧɧɨɟ ɜɵɪɚɠɟɧɢɟ
s |
∂L |
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Ɂɞɟɫɶ ɢɫɩɨɥɶɡɭɟɦ ɱɬɨδ qk t
t s |
ª |
∂L |
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³¦ |
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t k = |
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= δ qk t = δ qk = dtd δqk ɢ ɭɪɚɜɧɟɧɢɹ Ʌɚɝɪɚɧ
ɠɚ
ɋɩɪɚɜɟɞɥɢɜɨ ɢ ɨɛɪɚɬɧɨɟ ɭɬɜɟɪɠɞɟɧɢɟ ɢɡ ɩɪɢɧɰɢɩɚ ɷɤɫɬɪɟɦɚɥɶɧɨɝɨ ɞɟɣ
ɫɬɜɢɹ ɫɥɟɞɭɸɬ ɭɪɚɜɧɟɧɢɹ Ʌɚɝɪɚɧɠɚ
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ɅȿɄɐɂə ɈɋɇɈȼɕ ɆȿɏȺɇɂɄɂ ȽȺɆɂɅɖɌɈɇȺ
Ɉɛɨɛɳɟɧɧɚɹ ɦɟɯɚɧɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɫɢɫɬɟɦɵ
Ɋɚɫɫɦɨɬɪɢɦ ɩɨɥɧɭɸ ɩɪɨɢɡɜɨɞɧɭɸ ɩɨ ɜɪɟɦɟɧɢ ɨɬ ɮɭɧɤɰɢɢ Ʌɚɝɪɚɧɠɚ
dL |
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∂q |
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k= © |
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ȼ ɞɚɧɧɨɦ ɭɪɚɜɧɟɧɢɢ ɡɚɦɟɧɢɦ ∂L |
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∂qk |
ɱɟɪɟɡ |
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ɢɫɩɨɥɶɡɭɹ ɭɪɚɜɧɟɧɢɹ |
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Ʌɚɝɪɚɧɠɚ ɉɪɨɞɟɥɚɜ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ ɩɨɥɭɱɢɦ |
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ȼɟɥɢɱɢɧɚ ɫɬɨɹɳɚɹ ɜ ɫɤɨɛɤɚɯ ɮɨɪɦɭɥɵ |
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ɧɚɡɵɜɚɟɬɫɹ ɨɛɨɛɳɟɧɧɨɣ ɩɨɥɧɨɣ ɦɟɯɚɧɢɱɟɫɤɨɣ ɷɧɟɪɝɢɟɣ ɫɢɫɬɟɦɵ
ȼ ɬɨɦ ɫɥɭɱɚɟ ɤɨɝɞɚ ɧɚ ɫɢɫɬɟɦɭ ɧɚɤɥɚɞɵɜɚɸɬɫɹ ɫɬɚɰɢɨɧɚɪɧɵɟ ɫɜɹɡɢ ɞɥɹ
ɤɢɧɟɬɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ ɫɩɪɚɜɟɞɥɢɜɨ ɜɵɪɚɠɟɧɢɟ ɉɭɫɬɶ ɞɚɥɟɟ ɧɚ ɫɢɫɬɟɦɭ
ɞɟɣɫɬɜɭɸɬ ɬɨɥɶɤɨ ɩɨɬɟɧɰɢɚɥɶɧɵɟ ɫɢɥɵ ȼ ɷɬɨɦ ɫɥɭɱɚɟ
s |
∂L |
s |
∂T |
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¦qk |
= ¦qk |
= T |
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k = |
∂qk |
k = |
∂ qk |
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ɋɥɟɞɨɜɚɬɟɥɶɧɨ E = T − L = T − T − U = T + U Ɍɚɤɢɦ ɨɛɪɚɡɨɦ ɨɛɨɛɳɟɧ ɧɚɹ ɷɧɟɪɝɢɹ ɜ ɞɚɧɧɨɦ ɫɥɭɱɚɟ ɫɨɜɩɚɞɚɟɬ ɫ ɨɛɵɱɧɨɣ ɦɟɯɚɧɢɱɟɫɤɨɣ ɷɧɟɪɝɢɟɣ ɫɢɫɬɟɦɵ
Ɉɩɢɫɚɧɢɟ ɫɨɫɬɨɹɧɢɹ ɦɟɯɚɧɢɱɟɫɤɨɣ ɫɢɫɬɟɦɵ ɜ ɦɟɯɚɧɢɤɟ Ƚɚɦɢɥɶɬɨɧɚ
Ɏɚɡɨɜɨɟ ɩɪɨɫɬɪɚɧɫɬɜɨ
ȼ ɩɨɞɯɨɞɟ Ƚɚɦɢɥɶɬɨɧɚ ɫɨɫɬɨɹɧɢɟ ɦɟɯɚɧɢɱɟɫɤɨɣ ɫɢɫɬɟɦɵ ɨɩɢɫɵɜɚɟɬɫɹ ɫ ɩɨɦɨɳɶɸ ɨɛɨɛɳɟɧɧɵɯ ɤɨɨɪɞɢɧɚɬqk ɢ ɨɛɨɛɳɟɧɧɵɯ ɢɦɩɭɥɶɫɨɜ
pk = ∂ L k = s
∂qk
Ɏɨɪɦɭɥɵ ɫɥɭɠɚɬ ɨɩɪɟɞɟɥɟɧɢɟɦ ɨɛɨɛɳɟɧɧɵɯ ɢɦɩɭɥɶɫɨɜ Ɉɛɨɛɳɟɧ
ɧɵɟ ɤɨɨɪɞɢɧɚɬɵ ɢ ɢɦɩɭɥɶɫɵ ɱɚɫɬɢɰ ɩɨɥɧɨɫɬɶɸ ɨɩɪɟɞɟɥɹɸɬ ɫɨɫɬɨɹɧɢɟ ɫɢɫɬɟɦɵ
ɜ ɧɟɤɨɬɨɪɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ɢ ɜ ɩɨɫɥɟɞɭɸɳɢɟ ɦɨɦɟɧɬɵ ɜɪɟɦɟɧɢ Ɏɢɡɢɱɟɫɤɢɟ ɩɟɪɟɦɟɧɧɵɟɜ ɩɨɞɯɨɞɟȽɚɦɢɥɶɬɨɧɚɜɵɪɚɠɚɸɬɱɟɪɟɡɨɛɨɛɳɟɧɧɵɟɤɨɨɪɞɢɧɚɬɵ ɢ ɢɦɩɭɥɶɫɵ
57
М. А. Михайлов. ЛЕКЦИИ ПО КЛАССИЧЕСКОЙ МЕХАНИКЕ
ȼɜɟɞɟɦ ɩɨɧɹɬɢɟ ɮɚɡɨɜɨɝɨ ɩɪɨɫɬɪɚɧɫɬɜɚɎɚɡɨɜɵɦ ɩɪɨɫɬɪɚɧɫɬɜɨɦɧɚɡɵ ɜɚɟɬɫɹ s ɦɟɪɧɨɟ ɩɪɨɫɬɪɚɧɫɬɜɨ ɨɫɹɦɢ ɤɨɬɨɪɨɝɨ ɹɜɥɹɸɬɫɹ ɨɛɨɛɳɟɧɧɵɟ ɤɨɨɪ ɞɢɧɚɬɵ ɢ ɫɤɨɪɨɫɬɢ ɱɚɫɬɢɰ ɫɢɫɬɟɦɵ ɋɨɫɬɨɹɧɢɸ ɦɟɯɚɧɢɱɟɫɤɨɣ ɫɢɫɬɟɦɵ ɨɬɜɟ ɱɚɟɬ ɬɨɱɤɚ ɜ ɮɚɡɨɜɨɦ ɩɪɨɫɬɪɚɧɫɬɜɟ ɉɪɢ ɞɜɢɠɟɧɢɢ ɫɢɫɬɟɦɵ ɬɨɱɤɚ ɜ ɮɚɡɨɜɨɦ ɩɪɨɫɬɪɚɧɫɬɜɟ ɨɩɢɫɵɜɚɟɬ ɧɟɤɨɬɨɪɭɸ ɤɪɢɜɭɸ ɤɨɬɨɪɚɹ ɧɚɡɵɜɚɟɬɫɹ ɮɚɡɨɜɨɣ ɬɪɚɟɤɬɨɪɢɟɣ
Ɏɭɧɤɰɢɹ Ƚɚɦɢɥɶɬɨɧɚ Ʉɚɧɨɧɢɱɟɫɤɢɟ ɭɪɚɜɧɟɧɢɹ Ƚɚɦɢɥɶɬɨɧɚ
Ɏɭɧɤɰɢɟɣ Ƚɚɦɢɥɶɬɨɧɚɝɚɦɢɥɶɬɨɧɢɚɧɨɦ ɧɚɡɵɜɚɟɬɫɹ ɨɛɨɛɳɟɧɧɚɹ ɷɧɟɪɝɢɹ ɫɢɫɬɟɦɵ ɤɨɬɨɪɚɹ ɜɵɪɚɠɚɟɬɫɹ ɱɟɪɟɡ ɨɛɨɛɳɟɧɧɵɟ ɤɨɨɪɞɢɧɚɬɵ ɢ ɢɦɩɭɥɶɫɵ Ⱦɥɹ ɩɨɥɭɱɟɧɢɹ ɮɭɧɤɰɢɢ Ƚɚɦɢɥɶɬɨɧɚ ɫɥɟɞɭɟɬ ɜ ɮɨɪɦɭɥɟ ɩɟɪɟɣɬɢ ɨɬ ɨɛɨɛɳɟɧ ɧɵɯ ɫɤɨɪɨɫɬɟɣ ɤ ɨɛɨɛɳɟɧɧɵɦ ɢɦɩɭɥɶɫɚɦ ɂɬɚɤ ɩɨ ɨɩɪɟɞɟɥɟɧɢɸ ɮɭɧɤɰɢɹ Ƚɚ ɦɢɥɶɬɨɧɚ ɟɫɬɶ
s |
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H qk pk t = ¦pk qk |
−L qk pk t |
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k =
ɉɪɨɞɢɮɮɟɪɟɧɰɢɪɭɟɦ ɩɨ ɢɦɩɭɥɶɫɭ pk ȼ ɪɟɡɭɥɶɬɚɬɟ ɩɨɥɭɱɢɦ
∂ ɇ = qk − |
∂ L = qk |
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∂ pk |
∂ pk |
Ⱥɧɚɥɨɝɢɱɧɨ ɞɢɮɮɟɪɟɧɰɢɪɭɟɦ ɩɨ ɨɛɨɛɳɟɧɧɨɣ ɤɨɨɪɞɢɧɚɬɟ ɢ ɧɚɯɨɞɢɦ
∂ H |
= − |
∂L |
= − |
d |
(pk ) |
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dt |
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∂qk |
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Ɂɞɟɫɶ ɦɵ ɢɫɩɨɥɶɡɨɜɚɥɢ ɭɪɚɜɧɟɧɢɹ Ʌɚɝɪɚɧɠɚ ɢ ɨɩɪɟɞɟɥɟɧɢɟ ɨɛɨɛ ɳɟɧɧɨɝɨ ɢɦɩɭɥɶɫɚ ɉɨɫɥɟɞɧɢɟ ɞɜɚ ɭɪɚɜɧɟɧɢɹ ɧɚɡɵɜɚɸɬɫɹɤɚɧɨɧɢɱɟɫɤɢɦɢ ɭɪɚɜɧɟɧɢɹɦɢ Ƚɚɦɢɥɶɬɨɧɚɢ ɡɚɩɢɫɵɜɚɸɬɫɹ ɜ ɜɢɞɟ
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d qk |
= |
∂ H |
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d t |
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∂ pk |
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d pk |
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= − ∂ H |
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d t |
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∂ qk |
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ɉɟɪɟɦɟɧɧɵɟ qk pk ɤɨɬɨɪɵɟ ɜɯɨɞɹɬ ɜ ɤɚɧɨɧɢɱɟɫɤɢɟ ɭɪɚɜɧɟɧɢɹ Ƚɚɦɢɥɶɬɨ
ɧɚ ɧɚɡɵɜɚɸɬɫɹɤɚɧɨɧɢɱɟɫɤɢ ɫɨɩɪɹɠɟɧɧɵɦɢɩɟɪɟɦɟɧɧɵɦɢ ȿɫɥɢ ɮɭɧɤɰɢɹ Ƚɚɦɢɥɶɬɨɧɚ ɹɜɧɨ ɧɟ ɡɚɜɢɫɢɬ ɨɬ ɜɪɟɦɟɧɢ ɬɨ ɨɧɚ ɩɪɟɞɫɬɚɜ
ɥɹɟɬ ɫɨɛɨɣ ɩɨɥɧɭɸ ɦɟɯɚɧɢɱɟɫɤɭɸ ɷɧɟɪɝɢɸ ɫɢɫɬɟɦɵ ȼ ɨɛɳɟɦ ɫɥɭɱɚɟ ɨɧɚ ɩɪɟɞ ɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɨɛɨɛɳɟɧɧɭɸ ɷɧɟɪɝɢɸ ɜɵɪɚɠɟɧɧɭɸ ɱɟɪɟɡ ɨɛɨɛɳɟɧɧɵɟ ɤɨɨɪɞɢɧɚɬɵ ɢ ɫɤɨɪɨɫɬɢ
ɉɪɢɦɟɪɎɭɧɤɰɢɹ Ƚɚɦɢɥɶɬɨɧɚ ɞɥɹ ɱɚɫɬɢɰɵ ɜ ɩɨɥɟ ɫɢɥɵ ɬɹɠɟɫɬɢ ɢɦɟ ɟɬ ɜɢɞ
H = m (px + py + pz )+ m g z
58 |
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Лекция10.ОСНОВЫМЕХАНИКИГАМИЛЬТОНА
ɍɪɚɜɧɟɧɢɹ Ƚɚɦɢɥɶɬɨɧɚ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ
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= |
px |
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dy |
= |
py |
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dz |
= |
pz |
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dt |
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m |
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m |
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dpx |
= |
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dpz |
= −m g |
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ɋɤɨɛɤɢ ɉɭɚɫɫɨɧɚ
ɉɭɫɬɶɞɜɢɠɟɧɢɟɫɢɫɬɟɦɵɨɩɢɫɵɜɚɟɬɫɹɤɚɧɨɧɢɱɟɫɤɢɦɢɭɪɚɜɧɟɧɢɹɦɢȽɚ ɦɢɥɶɬɨɧɚ ± ɉɭɫɬɶ ɞɚɥɟɟ f qk pk t ɟɫɬɶ ɧɟɤɨɬɨɪɚɹ ɞɢɧɚɦɢɱɟɫɤɚɹ
ɮɭɧɤɰɢɹ ɨɛɨɛɳɟɧɧɵɯ ɤɨɨɪɞɢɧɚɬ ɢɦɩɭɥɶɫɨɜ ɜɪɟɦɟɧɢ ± ɷɧɟɪɝɢɹ ɢɦɩɭɥɶɫ ɦɨ
ɦɟɧɬ ɢɦɩɭɥɶɫɚ ɢ ɞɪ Ɂɚɩɢɲɟɦ ɩɨɥɧɭɸ ɩɪɨɢɡɜɨɞɧɭɸ ɩɨ ɜɪɟɦɟɧɢ ɨɬ ɮɭɧɤɰɢɢ f
d f |
s |
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∂ f |
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∂ f |
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dt |
=¦¨ |
∂q |
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pk ¸+ |
∂t |
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© |
k |
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k |
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¹ |
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k= |
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ɉɨɞɫɬɚɜɢɦ ɬɟɩɟɪɶ ɩɪɨɢɡɜɨɞɧɵɟ ɩɨ ɜɪɟɦɟɧɢ ɨɬ ɨɛɨɛɳɟɧɧɵɯ ɤɨɨɪɞɢɧɚɬ ɢ ɢɦɩɭɥɶɫɨɜ ɢɡ ɤɚɧɨɧɢɱɟɫɤɢɯ ɭɪɚɜɧɟɧɢɣ Ƚɚɦɢɥɶɬɨɧɚ ± ȼ ɪɟɡɭɥɶɬɚɬɟ ɩɨɥɭɱɢɦ
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ɝɞɟ |
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ɟɫɬɶ ɩɨ ɨɩɪɟɞɟɥɟɧɢɸɫɤɨɛɤɚ ɉɭɚɫɫɨɧɚɞɥɹ ɮɭɧɤɰɢɣ f H ɋɤɨɛɤɢ ɉɭɚɫɫɨɧɚ ɦɨɠɧɨ ɨɩɪɟɞɟɥɢɬɶ ɢ ɞɥɹ ɞɪɭɝɢɯ ɩɚɪ ɮɭɧɤɰɢɣ ɤɨ
ɬɨɪɵɟɡɚɜɢɫɹɬɨɬɨɛɨɛɳɟɧɧɵɯ ɤɨɨɪɞɢɧɚɬɨɛɨɛɳɟɧɧɵɯɢɦɩɭɥɶɫɨɜ ɢ ɜɪɟɦɟɧɢ ȼ ɬɨɦ ɱɢɫɥɟ ɫɤɨɛɤɢ ɉɭɚɫɫɨɧɚ ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɞɥɹ ɫɚɦɢɯ ɨɛɨɛɳɟɧɧɵɯ ɢɦɩɭɥɶ ɫɨɜ ɢ ɤɨɨɪɞɢɧɚɬ ɂɫɩɨɥɶɡɭɹ ɨɩɪɟɞɟɥɟɧɢɟ ɦɨɠɧɨ ɩɨɤɚɡɚɬɶ ɱɬɨ
{qk ql }= {pk pl }= {pk ql }= δkl
ɝɞɟ
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ɟɫɥɢ k ≠ l |
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ɋɤɨɛɤɢ ɉɭɚɫɫɨɧɚ ɧɚɡɵɜɚɸɬɮɭɧɞɚɦɟɧɬɚɥɶɧɵɦɢ ɫɤɨɛɤɚɦɢ ɉɭɚɫɫɨɧɚ ɋɤɨɛɤɢ ɉɭɚɫɫɨɧɚ ɨɛɥɚɞɚɸɬ ɫɥɟɞɭɸɳɢɦɢ ɫɜɨɣɫɬɜɚɦɢ
{λ F R}= λ{F R}
{F +F R}={F R}+{F R}{F F R}= F {F R}+{F R}F
59
М. А. Михайлов. ЛЕКЦИИ ПО КЛАССИЧЕСКОЙ МЕХАНИКЕ
{F R}= − {R F}
{F {F R}}+ {R {F F }}+ {F {R F }}=
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Ɂɚɤɨɧɵ ɫɨɯɪɚɧɟɧɢɹ ɜ ɦɟɯɚɧɢɤɟ Ʌɚɝɪɚɧɠɚ ɢ Ƚɚɦɢɥɶɬɨɧɚ |
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ɉɭɫɬɶɮɭɧɤɰɢɹɅɚɝɪɚɧɠɚ ɧɟɡɚɜɢɫɢɬɨɬɨɛɨɛɳɟɧɧɨɣɤɨɨɪɞɢɧɚɬɵ |
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qk |
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= ɌɨɝɞɚɢɡɭɪɚɜɧɟɧɢɣɅɚɝɪɚɧɠɚ ɢ ɨɩɪɟɞɟɥɟɧɢɹɨɛɨɛɳɟɧɧɨɝɨ |
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ɢɦɩɭɥɶɫɚ ɫɥɟɞɭɟɬ ɱɬɨ ɨɛɨɛɳɟɧɧɵɣ ɢɦɩɭɥɶɫ ɨɬɜɟɱɚɸɳɢɣ ɞɚɧɧɨɣ ɨɛɨɛ
ɳɟɧɧɨɣ ɤɨɨɪɞɢɧɚɬɟ ɫɨɯɪɚɧɹɟɬɫɹ Ɉɛɨɛɳɟɧɧɵɟ ɤɨɨɪɞɢɧɚɬɵ ɨɬ ɤɨɬɨɪɵɯ ɮɭɧɤ ɰɢɹ Ʌɚɝɪɚɧɠɚ ɧɟ ɡɚɜɢɫɢɬ ɧɚɡɵɜɚɸɬɫɹɰɢɤɥɢɱɟɫɤɢɦɢ
Ɍɪɟɛɨɜɚɧɢɟ ɨɛɪɚɳɟɧɢɹ ɜ ɧɭɥɶ ɱɚɫɬɧɨɣ ɩɪɨɢɡɜɨɞɧɨɣ ∂L
∂qk ɷɤɜɢɜɚɥɟɧɬɧɨ
ɬɪɟɛɨɜɚɧɢɸ ɱɬɨɛɵ ɮɭɧɤɰɢɹ Ʌɚɝɪɚɧɠɚ ɦɟɯɚɧɢɱɟɫɤɨɣ ɫɢɫɬɟɦɵ ɧɟ ɦɟɧɹɥɚɫɶ ɩɪɢ
ɩɪɟɨɛɪɚɡɨɜɚɧɢɢ
qk → qk + δ qk ql → ql k ≠ l
ɉɪɢ ɞɚɧɧɨɦ ɩɪɟɨɛɪɚɡɨɜɚɧɢɢ ɮɭɧɤɰɢɹ Ʌɚɝɪɚɧɠɚ ɩɨɥɭɱɚɟɬ ɩɪɢɪɚɳɟɧɢɟ
δL = ∂ L δ qk
∂qk
Ɉɬɫɸɞɚ ɫɥɟɞɭɟɬ ɱɬɨ ɟɫɥɢ ∂L
∂qk = ɬɨ ɩɪɢɪɚɳɟɧɢɟ ɮɭɧɤɰɢɢ Ʌɚɝɪɚɧɠɚ δL =
ɉɪɟɨɛɪɚɡɨɜɚɧɢɟ ɜ ɱɚɫɬɧɨɫɬɢ ɦɨɠɟɬ ɛɵɬɶ ɩɪɟɨɛɪɚɡɨɜɚɧɢɟɦ ɩɪɨ
ɫɬɪɚɧɫɬɜɟɧɧɨɝɨ ɫɞɜɢɝɚ ɢɥɢ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɨɝɨ ɩɨɜɨɪɨɬɚ ɨɛɨɛɳɟɧɧɚɹ ɤɨɨɪɞɢ
ɧɚɬɚ ɹɜɥɹɟɬɫɹ ɭɝɥɨɜɨɣ ɤɨɨɪɞɢɧɚɬɨɣ ɦɟɯɚɧɢɱɟɫɤɨɣ ɫɢɫɬɟɦɵ ɜɨɤɪɭɝ ɫɨɨɬɜɟɬɫɬ
ɜɭɸɳɟɣ ɨɫɢ ɜɪɚɳɟɧɢɹ ɍɫɥɨɜɢɹ ∂L = δL = ɜɵɪɚɠɚɸɬ ɩɪɢ ɷɬɨɦ ɡɚɤɨɧɵ ɫɨ
∂qk
ɯɪɚɧɟɧɢɹ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɯ ɩɪɨɟɤɰɢɣ ɢɦɩɭɥɶɫɚ ɢ ɦɨɦɟɧɬɚ ɢɦɩɭɥɶɫɚ
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ ɢɧɜɚɪɢɚɧɬɧɨɫɬɶ ɧɟɢɡɦɟɧɧɨɫɬɶ ɮɭɧɤɰɢɢ Ʌɚɝɪɚɧɠɚ ɦɟɯɚ
ɧɢɱɟɫɤɨɣ ɫɢɫɬɟɦɵ ɨɬɧɨɫɢɬɟɥɶɧɨ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ ɫɜɹɡɚɧɚ ɫ ɫɢɦɦɟɬɪɢɟɣ
ɜɧɟɲɧɟɝɨ ɫɢɥɨɜɨɝɨ ɩɨɥɹ ɢ ɧɚɥɨɠɟɧɧɵɯ ɧɚ ɫɢɫɬɟɦɭ ɫɜɹɡɟɣ
ɉɪɢɦɟɪȼ ɩɪɢɦɟɪɟ ɱɚɫɬɢɰɚ ɜ ɩɨɥɟ ɫɢɥɵ ɬɹɠɟɫɬɢ Ɂɟɦɥɢ ɮɭɧɤ ɰɢɹ Ʌɚɝɪɚɧɠɚ ɧɟ ɡɚɜɢɫɢɬ ɨɬ ɞɟɤɚɪɬɨɜɵɯ ɤɨɨɪɞɢɧɚɬx y ɢ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɩɪɨ ɟɤɰɢɢ ɢɦɩɭɥɶɫɚ ɱɚɫɬɢɰɵ ɫɨɯɪɚɧɹɸɬɫɹ ȿɫɥɢ ɜ ɤɚɱɟɫɬɜɟ ɨɛɨɛɳɟɧɧɵɯ ɤɨɨɪɞɢɧɚɬ ɢɫɩɨɥɶɡɨɜɚɬɶ ɰɢɥɢɧɞɪɢɱɟɫɤɢɟ ɤɨɨɪɞɢɧɚɬɵ ɬɨ ɮɭɧɤɰɢɹ Ʌɚɝɪɚɧɠɚ ɢɦɟɟɬ ɜɢɞ
L = |
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Ɏɭɧɤɰɢɹ Ʌɚɝɪɚɧɠɚ ɜ ɞɚɧɧɨɦ ɫɥɭɱɚɟ ɧɟ ɡɚɜɢɫɢɬ ɨɬ ɭɝɥɨɜɨɣ ɤɨɨɪɞɢɧɚɬɵϕ |
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ɢ ɨɛɨɛɳɟɧɧɵɣ ɢɦɩɭɥɶɫ∂L ∂ ϕ = mρ ϕ = const ± ɩɪɨɟɤɰɢɹ ɨɪɛɢɬɚɥɶɧɨɝɨ ɦɨɦɟɧɬɚ |
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ɧɚ ɨɫɶ z ɫɨɯɪɚɧɹɟɬɫɹ Ⱦɚɧɧɵɟ ɡɚɤɨɧɵ ɫɨɯɪɚɧɟɧɢɹ ɫɜɹɡɚɧɵ ɫ ɫɢɦɦɟɬɪɢɟɣ ɨɞɧɨ ɪɨɞɧɨɝɨ ɩɨɥɹ ɫɢɥɵ ɬɹɠɟɫɬɢ ɫɨɫɬɨɹɧɢɟ ɦɟɯɚɧɢɱɟɫɤɨɣ ɫɢɫɬɟɦɵ ɧɟ ɦɟɧɹɟɬɫɹ ɩɪɢ ɫɞɜɢɝɚɯ ɜɞɨɥɶ ɨɫɟɣx y ɢ ɩɪɢ ɩɨɜɨɪɨɬɟ ɜɨɤɪɭɝ ɨɫɢz
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