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Лекции по классической механике. Учебное пособие

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ɈɋɇɈȼɕ ɋɌɈ

ɅȿɄɐɂə ɉɈɋɌɍɅȺɌɕ ɋɌɈ ɂ ɂɏ ɋɅȿȾɋɌȼɂə

ɉɨɫɬɭɥɚɬɵ ɋɌɈ

ȼ ɨɫɧɨɜɟ ɩɨɫɬɪɨɟɧɢɹ ɤɥɚɫɫɢɱɟɫɤɨɣ ɦɟɯɚɧɢɤɢ ɥɟɠɢɬ ɩɪɢɧɰɢɩ ɨɬɧɨɫɢɬɟɥɶ

ɧɨɫɬɢ Ƚɚɥɢɥɟɹ ɡɚɤɨɧɵ ɦɟɯɚɧɢɤɢ ɞɨɥɠɧɵ ɛɵɬɶ ɤɨɜɚɪɢɚɧɬɧɵ ɨɬɧɨɫɢɬɟɥɶɧɨ ɩɟ

ɪɟɯɨɞɚ ɨɬ ɨɞɧɨɣ ɂɋɈ ɤ ɞɪɭɝɨɣ ɬ ɟ ɢɦɟɬɶ ɨɞɢɧɚɤɨɜɵɣ ɜɢɞ ɜ ɪɚɡɧɵɯ ɂɋɈ Ɏɢ

ɡɢɱɟɫɤɢɟ ɜɟɥɢɱɢɧɵ ɩɪɢ ɷɬɨɦ ɦɨɝɭɬ ɢɡɦɟɧɹɬɶɫɹ ɨɩɪɟɞɟɥɟɧɧɵɦ ɨɛɪɚɡɨɦ ɧɨ ɫɜɹɡɶ

ɦɟɠɞɭ ɧɢɦɢ ɜɵɪɚɠɚɟɦɚɹ ɡɚɤɨɧɨɦ ɦɟɯɚɧɢɤɢ ɨɫɬɚɟɬɫɹ ɧɟɢɡɦɟɧɧɨɣ Ʉɨɨɪɞɢɧɚɬɵ ɫɨɛɵɬɢɣ (x y z t)ɩɪɢ ɩɟɪɟɯɨɞɟ ɨɬ ɨɞɧɨɣ ɂɋɈ ɤ ɞɪɭɝɨɣ ɦɟ

ɧɹɸɬɫɹ ɜ ɫɨɝɥɚɫɢɢ ɫ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹɦɢ Ƚɚɥɢɥɟɹ

x′ = x y′ = y z′ = z − V t t′ = t

ɋɥɟɞɫɬɜɢɟɦ ɞɚɧɧɵɯ ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ ɹɜɥɹɟɬɫɹ ɤɥɚɫɫɢɱɟɫɤɢɣ ɡɚɤɨɧ ɫɥɨɠɟ

ɧɢɹ ɫɤɨɪɨɫɬɟɣ

v&′ = v& − V

ɢ ɧɟɢɡɦɟɧɧɨɫɬɶ ɢɧɜɚɪɢɚɧɬɧɨɫɬɶ ɭɫɤɨɪɟɧɢɹ

w&′ = w

ɉɨɫɥɟ ɩɨɫɬɪɨɟɧɢɹ ɷɥɟɤɬɪɨɞɢɧɚɦɢɤɢ Ɇɚɤɫɜɟɥɥɚ ɜɨɡɧɢɤ ɡɚɤɨɧɨɦɟɪɧɵɣ ɜɨ ɩɪɨɫ ɭɞɨɜɥɟɬɜɨɪɹɸɬ ɥɢ ɭɪɚɜɧɟɧɢɹ Ɇɚɤɫɜɟɥɥɚ ɩɪɢɧɰɢɩɭ ɨɬɧɨɫɢɬɟɥɶɧɨɫɬɢ ɍɪɚɜɧɟɧɢɹ Ɇɚɤɫɜɟɥɥɚ ɨɤɚɡɚɥɢɫɶ ɧɟɤɨɜɚɪɢɚɧɬɧɵ ɨɬɧɨɫɢɬɟɥɶɧɨ ɩɪɟɨɛɪɚɡɨɜɚ ɧɢɣ ȼ ɪɟɡɭɥɶɬɚɬɟ ɫɥɨɠɢɥɢɫɶ ɞɜɟ ɜɨɡɦɨɠɧɨɫɬɢ ɫɨɝɥɚɫɨɜɚɧɢɹ ɪɟɡɭɥɶɬɚɬɨɜ ɤɥɚɫɫɢɱɟɫɤɨɣ ɦɟɯɚɧɢɤɢ ɢ ɷɥɟɤɬɪɨɞɢɧɚɦɢɤɢ Ɉɬɤɚɡɚɬɶɫɹ ɨɬ ɩɪɢɧɰɢɩɚ ɨɬɧɨɫɢ ɬɟɥɶɧɨɫɬɢ ɢ ɨɫɬɚɜɢɬɶ ɧɟɢɡɦɟɧɧɵɦɢ ɭɪɚɜɧɟɧɢɹ ɷɥɟɤɬɪɨɞɢɧɚɦɢɤɢ ɉɪɢɧɰɢɩ ɨɬɧɨɫɢɬɟɥɶɧɨɫɬɢ ɨɫɬɚɟɬɫɹ ɧɟɢɡɦɟɧɧɵɦ ɚ ɮɨɪɦɚ ɭɪɚɜɧɟɧɢɣ Ɇɚɤɫɜɟɥɥɚ ɞɨɥɠɧɚ ɛɵɬɶ ɜɢɞɨɢɡɦɟɧɟɧɚ Ɉɬ ɜɬɨɪɨɝɨ ɩɪɟɞɩɨɥɨɠɟɧɢɹ ɪɚɡɭɦɧɨ ɫɪɚɡɭ ɨɬɤɚɡɚɬɶɫɹ ɜ ɫɢɥɭɬɨɝɨɱɬɨɭɪɚɜɧɟɧɢɹɷɥɟɤɬɪɨɞɢɧɚɦɢɤɢɩɪɚɜɢɥɶɧɨɨɩɢɫɵɜɚɸɬ ɨɝɪɨɦɧɨɟ ɱɢɫɥɨ ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɵɯ ɮɚɤɬɨɜ ɢ ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɵɯ ɹɜɥɟɧɢɣ Ɉɬɤɚɡɵɜɚɬɶɫɹ ɨɬ ɩɪɢɧɰɢɩɚ ɨɬɧɨɫɢɬɟɥɶɧɨɫɬɢ ɷɬɨ ɩɨ ɫɭɬɢ ɨɡɧɚɱɚɟɬ ɨɬɤɚɡɚɬɶɫɹ ɨɬ ɜɡɚɢɦɨɫɜɹɡɢ ɮɢɡɢɱɟɫɤɢɯ ɹɜɥɟɧɢɣ ɇɚ ɫɚɦɨɦ ɞɟɥɟ ɜ ɮɢɡɢɱɟɫɤɢɯ ɩɪɨɰɟɫɫɚɯ ɧɚɛɥɸɞɚɟɬɫɹ ɨɞ ɧɨɜɪɟɦɟɧɧɨɟɩɪɨɹɜɥɟɧɢɟɦɟɯɚɧɢɱɟɫɤɢɯɷɥɟɤɬɪɨɦɚɝɧɢɬɧɵɯɬɟɩɥɨɜɵɯɡɚɤɨɧɨ ɦɟɪɧɨɫɬɟɣ ɉɪɢɦɟɪɨɦ ɦɨɠɟɬ ɫɥɭɠɢɬɶ ɯɨɬɹ ɛɵ ɫɬɨɥɤɧɨɜɟɧɢɟ ɬɟɥ ɜ ɤɨɬɨɪɨɦ ɩɪɨɹɜɥɹɸɬɫɹ ɫɢɥɵ ɭɩɪɭɝɨɫɬɢ ɢɦɟɸɳɢɟ ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɭɸ ɩɪɢɪɨɞɭ Ɍɚɤɢɦ ɨɛ ɪɚɡɨɦ ɩɪɢɧɰɢɩ ɨɬɧɨɫɢɬɟɥɶɧɨɫɬɢ ɞɨɥɠɟɧ ɪɚɫɩɪɨɫɬɪɚɧɹɬɶɫɹ ɧɚ ɜɫɟ ɹɜɥɟɧɢɹ ɩɪɢ ɪɨɞɵ ɢɥɢ ɧɟ ɜɵɩɨɥɧɹɬɶɫɹ ɜɨɨɛɳɟ

Ⱦɥɹ ɪɟɲɟɧɢɹ ɞɚɧɧɨɣ ɡɚɞɚɱɢ ɛɵɥ ɜɵɛɪɚɧ ɬɪɟɬɢɣ ɩɭɬɶ ɪɟɚɥɢɡɭɟɦɵɣ Ⱥ ɗɣɧɲɬɟɣɧɨɦɉɪɢɧɰɢɩɨɬɧɨɫɢɬɟɥɶɧɨɫɬɢɢ ɭɪɚɜɧɟɧɢɹɷɥɟɤɬɪɨɞɢɧɚɦɢɤɢɨɫ ɬɚɸɬɫɹ ɧɟɢɡɦɟɧɧɵɦɢ ȼɢɞɨɢɡɦɟɧɢɬɶ ɫɥɟɞɭɟɬ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ Ƚɚɥɢɥɟɹ ɂɯ ɫɥɟ ɞɭɟɬ ɡɚɦɟɧɢɬɶ ɧɚ ɛɨɥɟɟ ɨɛɳɢɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ ɬɚɤɢɦɢ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹɦɢ ɹɜɢ ɥɢɫɶ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ Ʌɨɪɟɧɰɚ ɤɨɬɨɪɵɟ ɧɟ ɢɡɦɟɧɹɥɢ ɛɵ ɜɢɞ ɭɪɚɜɧɟɧɢɣ ɷɥɟɤɬɪɨɞɢɧɚɦɢɤɢ ɜ ɱɚɫɬɧɨɦ ɫɥɭɱɚɟ ɩɟɪɟɯɨɞɢɥɢ ɛɵ ɜ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ Ƚɚɥɢ ɥɟɹ ɉɪɢ ɷɬɨɦ ɫɥɟɞɭɟɬ ɩɟɪɟɫɦɨɬɪɟɬɶ ɫɜɨɣɫɬɜɚ ɩɪɨɫɬɪɚɧɫɬɜɚ ɜɪɟɦɟɧɢ ɜ ɱɚɫɬɧɨɫɬɢ ɱɟɬɜɟɪɬɨɟ ɪɚɜɟɧɫɬɜɨ ɜ ɮɨɪɦɭɥɚɯ

М. А. Михайлов. ЛЕКЦИИ ПО КЛАССИЧЕСКОЙ МЕХАНИКЕ

ȼ ɪɟɡɭɥɶɬɚɬɟ ɜɵɩɨɥɧɟɧɢɹ ɞɚɧɧɨɣ ɩɪɨɝɪɚɦɦɵ ɛɵɥɚ ɩɨɫɬɪɨɟɧɚ ɫɩɟɰɢɚɥɶɧɚɹ

ɬɟɨɪɢɹ ɨɬɧɨɫɢɬɟɥɶɧɨɫɬɢ ɋɌɈ

ȼ ɨɫɧɨɜɟ ɋɌɈ ɥɟɠɚɬ ɞɜɚ ɩɨɫɬɭɥɚɬɚɉɪɢɧɰɢɩɨɬɧɨɫɢɬɟɥɶɧɨɫɬɢɉɪɢɧɰɢɩɨɬɧɨɫɢɬɟɥɶɧɨɫɬɢɬɟɩɟɪɶɪɚɫɩɪɨ

ɫɬɪɚɧɹɟɬɫɹ ɧɚ ɜɫɟ ɮɢɡɢɱɟɫɤɢɟ ɡɚɤɨɧɵ Ɂɚɤɨɧɵ ɩɪɢɪɨɞɵ ɧɟ ɞɨɥɠɧɵ ɦɟɧɹɬɶ ɜɢɞ

ɩɪɢ ɩɟɪɟɯɨɞɟ ɨɬ ɨɞɧɨɣ ɂɋɈ ɤ ɞɪɭɝɨɣ

ɉɪɢɧɰɢɩ ɫɭɳɟɫɬɜɨɜɚɧɢɹ ɜ ɩɪɢɪɨɞɟ ɢɧɜɚɪɢɚɧɬɧɨɣ ɫɤɨɪɨɫɬɢ ȼ ɩɪɢɪɨɞɟ

ɫɭɳɟɫɬɜɭɟɬ ɤɨɧɟɱɧɚɹ ɫɤɨɪɨɫɬɶ ɩɟɪɟɞɚɱɢ ɜɡɚɢɦɨɞɟɣɫɬɜɢɣ ɤɨɬɨɪɚɹ ɧɟɢɡɦɟɧɧɚ ɜɨ ɜɫɟɯ ɂɋɈ Ɍɚɤɨɣ ɫɤɨɪɨɫɬɶɸ ɹɜɥɹɟɬɫɹ ɫɤɨɪɨɫɬɶ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɷɥɟɤɬɪɨɦɚɝ ɧɢɬɧɵɯ ɜɡɚɢɦɨɞɟɣɫɬɜɢɣ ɤɨɬɨɪɚɹ ɱɢɫɥɟɧɧɨ ɪɚɜɧɚ ɫɤɨɪɨɫɬɢ ɫɜɟɬɚ ɜ ɜɚɤɭɭɦɟ

ɉɟɪɜɵɣ ɩɨɫɬɭɥɚɬ ɹɜɥɹɟɬɫɹ ɨɛɨɛɳɟɧɢɟɦ ɦɟɯɚɧɢɱɟɫɤɨɝɨ ɩɪɢɧɰɢɩɚ ɨɬɧɨɫɢ

ɬɟɥɶɧɨɫɬɢ ȼɬɨɪɨɣ ɩɨɫɬɭɥɚɬ ɩɨ ɫɭɬɢ ɜɵɪɚɠɚɟɬ ɝɢɩɨɬɟɡɭ ɨ ɬɨɦ ɱɬɨ ɢɧɜɚɪɢɚɧɬ

ɧɚɹ ɫɤɨɪɨɫɬɶ ɢɦɟɟɬ ɭɧɢɜɟɪɫɚɥɶɧɵɣ ɯɚɪɚɤɬɟɪ ɧɟ ɫɜɹɡɚɧɚ ɫ ɮɢɡɢɱɟɫɤɨɣ ɩɪɢɪɨɞɨɣ ɜɡɚɢɦɨɞɟɣɫɬɜɢɣ ɝɪɚɜɢɬɚɰɢɨɧɧɨɟ ɜɡɚɢɦɨɞɟɣɫɬɜɢɟ ɬɚɤɠɟ ɪɚɫɩɪɨɫɬɪɚɧɹɟɬɫɹ ɫɨ ɫɤɨɪɨɫɬɶɸ ɫɜɟɬɚ ɚ ɫɜɹɡɚɧɚ ɫɨ ɫɜɨɣɫɬɜɚɦɢ ɩɪɨɫɬɪɚɧɫɬɜɚ ɢ ɜɪɟɦɟɧɢ

ɉɪɟɨɛɪɚɡɨɜɚɧɢɹ Ʌɨɪɟɧɰɚ

Ɋɚɫɫɦɨɬɪɢɦ ɤɚɤ ɩɪɟɨɛɪɚɡɭɸɬɫɹ ɤɨɨɪɞɢɧɚɬɵ ɫɨɛɵɬɢɣ ɩɪɢ ɩɟɪɟɯɨɞɟ ɨɬ ɨɞ ɧɨɣ ɂɋɈ ɤ ɞɪɭɝɨɣ ɉɪɟɨɛɪɚɡɨɜɚɧɢɹ ± ɤɨɬɨɪɵɟ ɛɵɥɢ ɪɚɫɫɦɨɬɪɟɧɵ ɜ ɤɥɚɫɫɢɱɟɫɤɨɣ ɦɟɯɚɧɢɤɟ ɨɫɬɚɸɬɫɹ ɧɟɢɡɦɟɧɧɵɦɢ ɉɪɟɨɛɪɚɡɨɜɚɧɢɹ Ƚɚɥɢɥɟɹ ɡɚɦɟɧɹɸɬɫɹ ɧɚ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ Ʌɨɪɟɧɰɚ ɉɭɫɬɶ x y z t ɢx′ y′ z′ t′ ɤɨɨɪ ɞɢɧɚɬɵ ɫɨɛɵɬɢɹ ɜ ɂɋɈK K′ ɤɨɬɨɪɵɟ ɞɜɢɠɭɬɫɹ ɨɬɧɨɫɢɬɟɥɶɧɨ ɞɪɭɝ ɞɪɭɝɚ ɜɞɨɥɶ ɨɫɢz ɫ ɩɨɫɬɨɹɧɧɨɣ ɫɤɨɪɨɫɬɶɸV ɪɢɫ

K K′

y y′

x x′

z z′

Ɋɢɫ. 13.1.Ⱦɜɟ ɂɋɈ

ɉɭɫɬɶɜ ɦɨɦɟɧɬɜɪɟɦɟɧɢ t = ɧɚɱɚɥɚɫɢɫɬɟɦɤɨɨɪɞɢɧɚɬɫɨɜɩɚɞɚɸɬɢ ɢɡ ɧɚɱɚɥɚ ɫɢɫɬɟɦ ɤɨɨɪɞɢɧɚɬ ɪɚɫɩɪɨɫɬɪɚɧɹɟɬɫɹ ɫɮɟɪɢɱɟɫɤɚɹ ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɚɹ ɜɨɥ ɧɚ ɍɪɚɜɧɟɧɢɟ ɟɟ ɜɨɥɧɨɜɨɣ ɩɨɜɟɪɯɧɨɫɬɢ

x + y + z − c t =

ɂɫɩɨɥɶɡɭɹɢɧɜɚɪɢɚɧɬɧɨɫɬɶɫɤɨɪɨɫɬɢɫɜɟɬɚɢ ɧɟɢɡɦɟɧɧɨɫɬɶɡɚɤɨɧɚ ɩɪɢ ɩɟɪɟɯɨɞɟ ɤ ɫɢɫɬɟɦɟ K′ ɭɪɚɜɧɟɧɢɟ ɫɮɟɪɢɱɟɫɤɨɣ ɜɨɥɧɵ ɜ ɫɢɫɬɟɦɟ K′ ɡɚɩɢ ɲɟɦ ɜ ɜɢɞɟ

(x′) + (y′) + (z′) − c (t′) =

72

Лекция13.ПОСТУЛАТЫСТОИИХСЛЕДСТВИЯ

Ɏɨɪɦɭɥɵ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ ɤɨɨɪɞɢɧɚɬ ɫɨɛɵɬɢɹ ɞɨɥɠɧɵ ɧɟ ɧɚɪɭɲɚɬɶ ɫɨɨɬ

ɧɨɲɟɧɢɣ ɢ ɢ ɛɵɬɶ ɥɢɧɟɣɧɵɦɢ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹɦɢ ɍɫɥɨɜɢɟ ɥɢɧɟɣ

ɧɨɫɬɢ ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ ɫɥɟɞɭɟɬ ɢɡ ɫɜɨɣɫɬɜɚ ɨɞɧɨɪɨɞɧɨɫɬɢ ɩɪɨɫɬɪɚɧɫɬɜɚ Ɍɚɤɢɦɢ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹɦɢ ɩɪɢ ɜɵɛɨɪɟ ɫɢɫɬɟɦ ɨɬɫɱɟɬɚ ɧɚ ɪɢɫ

ɹɜɥɹɸɬɫɹ

x′ = x y′ = y z′ = α (V )(z − V t) t′ = β t + γ z

Ɂɞɟɫɶ ɤɨɷɮɮɢɰɢɟɧɬ α (V )ɡɚɜɢɫɢɬ ɨɬ ɫɤɨɪɨɫɬɢ ɨɬɧɨɫɢɬɟɥɶɧɨɝɨ ɞɜɢɠɟɧɢɹ ɫɢɫɬɟɦ ɨɬɫɱɟɬɚ ɤɨɷɮɮɢɰɢɟɧɬɵ β γ ɜ ɨɛɳɟɦ ɫɥɭɱɚɟ ɡɚɜɢɫɹɬ ɨɬ ɫɤɨɪɨɫɬɢ V ȼ ɱɚɫɬɧɨɦ ɫɥɭɱɚɟα = β = γ = ɦɵ ɩɨɥɭɱɚɟɦ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ Ƚɚɥɢɥɟɹ ɉɨɞ ɫɬɚɜɥɹɹ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ ɜ ɮɨɪɦɭɥɭ ɩɪɢɪɚɜɧɢɜɚɹ ɪɟɡɭɥɶɬɚɬ ɩɨɞɫɬɚ ɧɨɜɤɢ ɤ ɥɟɜɨɣ ɱɚɫɬɢ ɫɨɨɬɧɨɲɟɧɢɹ ɩɨɥɭɱɢɦ

x + y + α (z − V t ) − c (β t + γ z) = x + y + z − c t

Ⱦɚɧɧɨɟ ɪɚɜɟɧɫɬɜɨ ɛɭɞɟɬ ɜɵɩɨɥɧɟɧɨ ɟɫɥɢ

α − c γ = α V − c β = −c α V + c β γ =

ɂɡ ɞɚɧɧɵɯ ɭɪɚɜɧɟɧɢɣ ɧɚɯɨɞɢɦ

α = β =

γ = −

− V

ɉɨɞɫɬɚɜɥɹɹ ɧɚɣɞɟɧɧɵɟ ɡɧɚɱɟɧɢɹ ɜ ɮɨɪɦɭɥɵ ɧɚɣɞɟɦ

z − V t

t −

x′ = x y′ = y z′ =

t′ =

− V

Ⱦɚɧɧɵɟ ɮɨɪɦɭɥɵ ɹɜɥɹɸɬɫɹ ɢɫɤɨɦɵɦɢɩɪɟɨɛɪɚɡɨɜɚɧɢɹɦɢ Ʌɨɪɟɧɰɚɤɨɬɨ ɪɵɟ ɡɚɦɟɧɹɸɬ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ Ƚɚɥɢɥɟɹ

ɉɪɢ ɦɚɥɵɯ ɫɤɨɪɨɫɬɹɯ V << c ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ Ʌɨɪɟɧɰɚ ɩɟɪɟɯɨɞɹɬ ɜ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹȽɚɥɢɥɟɹȼ ɷɬɨɦɧɚɯɨɞɢɬɨɬɪɚɠɟɧɢɟ ɩɪɢɧɰɢɩ ɫɨɨɬɜɟɬɫɬɜɢɹ ɛɨɥɟɟ ɨɛɳɚɹ ɬɟɨɪɢɹ ɫɨɞɟɪɠɢɬ ɜ ɫɟɛɟ ɜ ɤɚɱɟɫɬɜɟ ɩɪɟɞɟɥɶɧɨɝɨ ɫɥɭɱɚɹ ɦɟɧɟɟ ɨɛ ɳɭɸ ɬɟɨɪɢɸ ɋɌɈ ɫɨɞɟɪɠɢɬ ɜ ɫɟɛɟ ɜ ɤɚɱɟɫɬɜɟ ɩɪɟɞɟɥɶɧɨɝɨ ɫɥɭɱɚɹ ɦɚɥɵɯ ɫɤɨ ɪɨɫɬɟɣ ɤɥɚɫɫɢɱɟɫɤɭɸ ɦɟɯɚɧɢɤɭ ɇɶɸɬɨɧɚ ± Ƚɚɥɢɥɟɹ ɉɪɢ ɫɤɨɪɨɫɬɹɯv << c ɫɨɨɬ ɧɨɲɟɧɢɹ ɋɌɈ ɞɨɥɠɧɵ ɩɟɪɟɯɨɞɢɬɶ ɜ ɫɨɨɬɧɨɲɟɧɢɹ ɤɥɚɫɫɢɱɟɫɤɨɣ ɦɟɯɚɧɢɤɢ

Ɂɚɩɢɲɟɦ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ Ʌɨɪɟɧɰɚ ɞɥɹ ɫɥɭɱɚɹ ɩɪɨɢɡɜɨɥɶɧɨɝɨ ɧɚ ɩɪɚɜɥɟɧɢɹ ɫɤɨɪɨɫɬɢV

r& − V t

[[r& V

]V ]

t −

& &

r V

r′ =

+ ¨

− ¸

t′ =

−

М. А. Михайлов. ЛЕКЦИИ ПО КЛАССИЧЕСКОЙ МЕХАНИКЕ

ɉɪɨɫɬɪɚɧɫɬɜɟɧɧɵɟ ɢ ɜɪɟɦɟɧɧɵɟ ɩɪɨɦɟɠɭɬɤɢ ɜ ɋɌɈ

ȼ ɫɨɝɥɚɫɢɢ ɫ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹɦɢ Ƚɚɥɢɥɟɹ ɢ ɨɫɧɨɜɧɵɦɢ ɩɨɥɨɠɟɧɢɹɦɢ ɤɥɚɫ

ɫɢɱɟɫɤɨɣ ɦɟɯɚɧɢɤɢ ɞɥɢɧɵ ɨɬɪɟɡɤɨɜ ɢ ɩɪɨɦɟɠɭɬɤɢ ɜɪɟɦɟɧɢ ɦɟɠɞɭ ɫɨɛɵɬɢɹɦɢ

ɨɞɢɧɚɤɨɜɵ ɜ ɪɚɡɧɵɯ ɂɋɈ

(x′ − x′ ) + (y′ − y′ ) + (z′ − z′ ) = (x − x ) + (y − y ) + (z − z )

t′ − t′ = t − t

Ɂɞɟɫɶ(x′ y′ z′ t′) (x y z t )± ɤɨɨɪɞɢɧɚɬɵ ɩɟɪɜɨɝɨ ɫɨɛɵɬɢɹ ɜ ɫɢɫɬɟɦɚɯK′ K(x′ y′ z′ t′ ) (x y z t )± ɤɨɨɪɞɢɧɚɬɵ ɜɬɨɪɨɝɨ ɫɨɛɵɬɢɹ ɜ ɫɢɫɬɟɦɚɯ K′ K ɉɪɨɰɟ ɞɭɪɚɢɡɦɟɪɟɧɢɹɞɥɢɧɦɚɫɲɬɚɛɨɜɩɪɨɫɬɪɚɧɫɬɜɟɧɧɵɯɩɪɨɦɟɠɭɬɤɨɜɦɟɠɞɭɫɨ ɛɵɬɢɹɦɢɢ ɩɪɨɦɟɠɭɬɤɨɜɜɪɟɦɟɧɢɜ ɤɥɚɫɫɢɱɟɫɤɨɣɦɟɯɚɧɢɤɟɯɨɪɨɲɨɢɡɜɟɫɬɧɚ ɂɡɦɟɪɟɧɢɟ ɞɥɢɧɵ ɫɜɨɞɢɬɫɹ ɤ ɫɪɚɜɧɟɧɢɸ ɟɟ ɫ ɷɬɚɥɨɧɨɦ ɞɥɢɧɵ ɤɨɬɨɪɨɟ ɨɫɭɳɟ ɫɬɜɥɹɟɬɫɹ ©ɩɪɢɤɥɚɞɵɜɚɧɢɟɦª ɷɬɚɥɨɧɚ ɤ ɢɡɦɟɪɹɟɦɨɦɭ ɦɚɫɲɬɚɛɭ Ⱦɥɢɧɚ ɬɟɥɚ ɧɟ ɡɚɜɢɫɢɬ ɨɬ ɟɝɨ ɫɤɨɪɨɫɬɢ ɞɜɢɠɟɧɢɹ ɢ ɞɚɧɧɭɸ ɩɪɨɰɟɞɭɪɭ ɢɡɦɟɪɟɧɢɹ ɞɨɫɬɚɬɨɱɧɨ ɩɪɨɜɟɫɬɢɞɥɹɧɟɩɨɞɜɢɠɧɨɝɨɦɚɫɲɬɚɛɚȾɥɹɢɡɦɟɪɟɧɢɹɩɪɨɦɟɠɭɬɤɨɜɜɪɟɦɟɧɢ ɞɨɫɬɚɬɨɱɧɨ ɨɞɧɢɯ ɱɚɫɨɜ ɤɨɬɨɪɵɟ ɨɛɵɱɧɨ ɫɱɢɬɚɸɬ ɧɚɯɨɞɹɳɢɦɢɫɹ ɜ ɧɚɱɚɥɟ ɫɢɫ ɬɟɦɵ ɤɨɨɪɞɢɧɚɬ

ɂɧɚɹ ɫɢɬɭɚɰɢɹ ɜɨɡɧɢɤɚɟɬ ɜ ɋɌɈ Ɂɞɟɫɶ ɞɥɢɧɵ ɨɬɪɟɡɤɨɜ ɩɪɨɦɟɠɭɬɤɢ ɦɟɠɞɭ ɫɨɛɵɬɢɹɦɢ ɨɤɚɡɵɜɚɸɬɫɹ ɡɚɜɢɫɹɳɢɦɢ ɨɬ ɢɯ ɫɤɨɪɨɫɬɢ ɫɢɫɬɟɦɵ ɨɬɫɱɟ ɬɚɜ ɤɨɬɨɪɨɣɩɪɨɜɨɞɢɬɫɹɢɡɦɟɪɟɧɢɟɩɪɨɫɬɪɚɧɫɬɜɟɧɧɨɝɨɩɪɨɦɟɠɭɬɤɚɦɟɠɞɭ ɫɨɛɵɬɢɹɦɢ ȼɪɟɦɟɧɧɵɟ ɩɪɨɦɟɠɭɬɤɢ ɦɟɠɞɭ ɫɨɛɵɬɢɹɦɢ ɬɚɤɠɟ ɡɚɜɢɫɹɬ ɨɬ ɜɵ ɛɨɪɚ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ

Ⱦɥɢɧɭ ɦɚɫɲɬɚɛɚ ɜ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ ɜ ɤɨɬɨɪɨɣ ɨɧ ɩɨɤɨɢɬɫɹ ɧɚɡɵɜɚɸɬɫɨɛ ɫɬɜɟɧɧɨɣ ɞɥɢɧɨɣɉɭɫɬɶ ɦɚɫɲɬɚɛ ɩɨɤɨɢɬɫɹ ɜ ɫɢɫɬɟɦɟ K′ ȿɝɨ ɤɨɨɪɞɢɧɚɬɵ z′ z′ ɢ ɫɨɛɫɬɜɟɧɧɚɹ ɞɥɢɧɚl = z′ − z′ ɉɭɫɬɶ ɦɚɫɲɬɚɛ ɞɜɢɠɟɬɫɹ ɨɬɧɨɫɢɬɟɥɶɧɨ ɫɢɫɬɟɦɵ

ɨɬɫɱɟɬɚ K ɫɨ ɫɤɨɪɨɫɬɶɸv ɜɞɨɥɶ ɨɫɢ z Ʉɨɧɤɪɟɬɢɡɢɪɭɟɦ ɩɪɨɰɟɞɭɪɭ ɢɡɦɟɪɟɧɢɹ ɞɜɢɠɭɳɟɝɨɫɹ ɦɚɫɲɬɚɛɚ Ⱦɥɹ ɟɝɨ ɢɡɦɟɪɟɧɢɹ ɧɟɨɛɯɨɞɢɦɨ ɡɚɮɢɤɫɢɪɨɜɚɬɶ ɤɨɨɪɞɢ ɧɚɬɵ ɧɚɱɚɥɚ ɢ ɤɨɧɰɚ ɜ ɨɞɢɧ ɢ ɬɨɬ ɠɟ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ Ɍɚɤɭɸ ɩɪɨɰɟɞɭɪɭ ɦɨɠɧɨ ɨɫɭɳɟɫɬɜɢɬɶ ɫ ɩɨɦɨɳɶɸ ɫɜɟɬɨɜɨɝɨ ɫɢɝɧɚɥɚ ɢɫɯɨɞɹɳɟɝɨ ɢɡ ɫɢɫɬɟɦɵ K′ ɉɭɫɬɶ ɜ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ t ɤɨɨɪɞɢɧɚɬɵ ɧɚɱɚɥɚ ɢ ɤɨɧɰɚ ɦɚɫɲɬɚɛɚ ɟɫɬɶ z z Ɍɨɝɞɚ ɢɡ ɬɪɟɬɶɟɣ ɮɨɪɦɭɥɵ ɧɚɯɨɞɢɦ

z′ =	z − vt	z′ =		z − vt

	− v			− v

	c				c
	z′ − z′ =		z − z

			− v

				c

74

Лекция13.ПОСТУЛАТЫСТОИИХСЛЕДСТВИЯ

Ɉɛɨɡɧɚɱɢɦ ɞɥɢɧɭ ɞɜɢɠɭɳɟɝɨɫɹ ɫɨ ɫɤɨɪɨɫɬɶɸv ɦɚɫɲɬɚɛɚ ɱɟɪɟɡl = z − z Ɉɤɨɧɱɚɬɟɥɶɧɨ ɩɨɥɭɱɢɦ

		v
l = l	−	v
l = l	−	c
		c

Ɍɚɤɢɦ ɨɛɪɚɡɨɦ ɞɥɢɧɚ ɦɚɫɲɬɚɛɚ ɞɜɢɠɭɳɟɝɨɫɹ ɫɨ ɫɤɨɪɨɫɬɶɸv ɫɨɤɪɚɳɚ ɟɬɫɹ ɜ ɫɨɝɥɚɫɢɢ ɫ ɮɨɪɦɭɥɨɣ ɗɬɨ ɫɨɤɪɚɳɟɧɢɟ ɪɚɡɦɟɪɨɜ ɩɪɨɢɫɯɨɞɢɬ ɜ ɧɚɩɪɚɜɥɟɧɢɢ ɫɤɨɪɨɫɬɢv ȼ ɧɚɩɪɚɜɥɟɧɢɢ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨɦ ɤ ɫɤɨɪɨɫɬɢ ɞɥɢ ɧɚ ɦɚɫɲɬɚɛɚ ɧɟ ɢɡɦɟɧɹɟɬɫɹ ɉɨɷɬɨɦɭ ɨɛɴɟɦ ɞɜɢɠɭɳɟɝɨɫɹ ɬɟɥɚ ɫɜɹɡɚɧ ɫ ɟɝɨ ɫɨɛ ɫɬɜɟɧɧɵɦ ɨɛɴɟɦɨɦ ɮɨɪɦɭɥɨɣ


V = V	−	v

		c

Ɍɚɤɢɦ ɨɛɪɚɡɨɦ ɪɚɫɫɬɨɹɧɢɟ ɦɟɠɞɭ ɞɜɭɦɹ ɬɨɱɤɚɦɢ ɩɪɨɫɬɪɚɧɫɬɜɚ ɞɥɢɧɚ ɢ ɨɛɴɟɦ ɦɚɫɲɬɚɛɚ ɡɚɜɢɫɢɬ ɨɬ ɜɵɛɨɪɚ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ ɬ ɟ ɧɨɫɢɬ ɨɬɧɨɫɢɬɟɥɶ ɧɵɣ ɯɚɪɚɤɬɟɪ

Ɋɚɫɫɦɨɬɪɢɦ ɤɚɤ ɨɛɫɬɨɢɬ ɞɟɥɨ ɫ ɩɪɨɦɟɠɭɬɤɚɦɢ ɜɪɟɦɟɧɢ ɉɭɫɬɶ ɜ ɫɢɫɬɟɦɟ K′ ɞɜɚ ɫɨɛɵɬɢɹ ɩɪɨɢɫɯɨɞɹɬ ɜ ɨɞɧɨɣ ɬɨɱɤɟ ɩɪɨɫɬɪɚɧɫɬɜɚ z′ = z′ = z′ ɨɞɧɨɦɟɫɬ ɧɵɟ ɫɨɛɵɬɢɹ ɉɪɨɦɟɠɭɬɨɤ ɜɪɟɦɟɧɢ ɦɟɠɞɭ ɧɢɦɢ

t = t′ − t′

ȼɫɢɫɬɟɦɟɨɬɫɱɟɬɚ K ɞɥɹɦɨɦɟɧɬɨɜɷɬɢɯɫɨɛɵɬɢɣɧɚɣɞɟɦɜɱɟɬɜɟɪɬɨɣ ɮɨɪɦɭɥɟ ɫɥɟɞɭɟɬ ɫɞɟɥɚɬɶ ɡɚɦɟɧɭ v → −v ɢ ɧɟɲɬɪɢɯɨɜɚɧɧɵɟ ɜɟɥɢɱɢɧɵ ɡɚ ɦɟɧɢɬɶ ɲɬɪɢɯɨɜɚɧɧɵɦɢ

	t′ +	v z′		t′ +	v z	′

t =		c	t =		c

	− v			− v

		c			c

ȼɜɨɞɹ ɩɪɨɦɟɠɭɬɨɤ ɜɪɟɦɟɧɢ t = t − t ɦɟɠɞɭ ɞɜɭɦɹ ɞɚɧɧɵɦɢ ɫɨɛɵɬɢɹɦɢ ɜ ɫɢɫɬɟɦɟK ɧɚɣɞɟɦ

t	=	t

− v

ȼɪɟɦɹ ɦɟɠɞɭ ɞɜɭɦɹ ɨɞɧɨɦɟɫɬɧɵɦɢ ɫɨɛɵɬɢɹɦɢ t ɜ ɞɚɧɧɨɣ ɫɢɫɬɟɦɟ ɨɬ

ɫɱɟɬɚ ɞɜɚ ɫɨɛɵɬɢɹ ɩɪɨɢɫɯɨɞɹɬ ɜ ɨɞɧɨɣ ɬɨɱɤɟ ɩɪɨɫɬɪɚɧɫɬɜɚ ɧɨ ɜ ɪɚɡɧɨɟ ɜɪɟɦɹ ɧɚɡɵɜɚɟɬɫɹɫɨɛɫɬɜɟɧɧɵɦ ɜɪɟɦɟɧɟɦ

ȼɚɠɧɵɦ ɜɵɜɨɞɨɦ ɢɡ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ ɹɜɥɹɟɬɫɹ ɫɥɟɞɭɸɳɢɣ ɯɨɞ ɱɚ ɫɨɜɜ ɪɚɡɥɢɱɧɵɯɂɋɈɨɤɚɡɵɜɚɟɬɫɹɧɟɨɞɢɧɚɤɨɜɵɦȾɜɢɠɭɳɢɟɫɹɨɬɧɨɫɢɬɟɥɶɧɨ ɧɟɤɨɬɨɪɨɣ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ ɱɚɫɵ ɢɞɭɬ ɦɟɞɥɟɧɧɟɟ ɱɟɦ ɱɚɫɵ ɤɨɬɨɪɵɟ ɩɨɤɨɹɬɫɹ ɜ ɷɬɨɣ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ

ɉɪɨɦɟɠɭɬɨɤ ɜɪɟɦɟɧɢ ɦɟɠɞɭ ɫɨɛɵɬɢɹɦɢ ɨɤɚɡɵɜɚɟɬɫɹ ɨɬɧɨɫɢɬɟɥɶɧɵɦ Ɋɚɫɫɦɨɬɪɢɦ ɫ ɞɚɧɧɵɯ ɩɨɡɢɰɢɣ ɩɨɫɬɪɨɟɧɢɟ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ ɜ ɋɌɈ

ȼ ɤɥɚɫɫɢɱɟɫɤɨɣ ɦɟɯɚɧɢɤɟ ɞɥɹ ɩɨɫɬɪɨɟɧɢɹ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ ɞɨɫɬɚɬɨɱɧɨ ɧɚɥɢɱɢɹ

ɨɞɧɢɯ ɱɚɫɨɜ ɤɨɬɨɪɵɟ ɩɨɦɟɳɚɸɬ ɜ ɧɚɱɚɥɨ ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ ȼ ɋɌɈ ɫɭɳɟɫɬ

М. А. Михайлов. ЛЕКЦИИ ПО КЛАССИЧЕСКОЙ МЕХАНИКЕ

ɜɨɜɚɧɢɟ ɤɨɧɟɱɧɨɣ ɫɤɨɪɨɫɬɢ ɩɟɪɟɞɚɱɢ ɜɡɚɢɦɨɞɟɣɫɬɜɢɣ ɩɪɢɜɨɞɢɬ ɤ ɧɟɨɛɯɨɞɢɦɨɫɬɢ ɧɚɥɢɱɢɹ ɧɚɛɨɪɚ ɱɚɫɨɜ ɤɨɬɨɪɵɟ ɞɨɥɠɧɵ ɛɵɬɶ ɩɨɦɟɳɟɧɵ ɜ ɤɚɠɞɨɣ ɬɨɱɤɟ ɩɪɨɫɬɪɚɧɫɬɜɚ ȼɫɟ ɱɚɫɵ ɞɚɧɧɨɣ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ ɞɨɥɠɧɵ ɛɵɬɶ ɧɟɩɨɞɜɢɠɧɵ ɜ ɞɚɧɧɨɣ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ ɉɪɢ ɷɬɨɦ ɯɨɞ ɱɚɫɨɜ ɞɨɥɠɟɧ ɛɵɬɶ ɫɢɧ ɯɪɨɧɢɡɢɪɨɜɚɧ ɉɪɨɰɟɞɭɪɚ ɫɢɧɯɪɨɧɢɡɚɰɢɢ ɱɚɫɨɜ ɦɨɠɟɬ ɛɵɬɶ ɭɫɬɚɧɨɜɥɟɧɚ ɫ ɩɨɦɨɳɶɸ ɫɜɟɬɨɜɵɯ ɫɢɝɧɚɥɨɜ ɪɢɫ ɑɚɫɵ ɪɚɡɜɟɞɟɧɵ ɧɚ ɨɞɢɧɚɤɨɜɵɟ ɪɚɫ ɫɬɨɹɧɢɹ ɨɬ ɬɨɱɤɢ ɝɞɟ ɧɚɯɨɞɢɬɫɹ ɢɫɬɨɱɧɢɤ ɫɜɟɬɨɜɨɝɨ ɫɢɝɧɚɥɚ ȼ ɦɨɦɟɧɬ ɩɪɢɯɨɞɚ ɫɜɟɬɨɜɵɯ ɫɢɝɧɚɥɨɜ ɜ ɬɨɱɤɢ ɪɚɫɩɨɥɨɠɟɧɢɹ ɱɚɫɨɜ ɱɚɫɵ ɞɨɥɠɧɵ ɩɨɤɚɡɵɜɚɬɶ ɨɞɧɨ ɢ ɬɨ ɠɟ ɡɧɚɱɟɧɢɟ ɦɨɦɟɧɬɚ ɜɪɟɦɟɧɢ

Ɋɢɫ. 13.2.ɉɪɨɰɟɞɭɪɚ ɫɢɧɯɪɨɧɢɡɚɰɢɢ ɱɚɫɨɜ

ɉɨɫɬɪɨɟɧɢɟɩɪɨɫɬɪɚɧɫɬɜɟɧɧɨɣɫɢɫɬɟɦɵɤɨɨɪɞɢɧɚɬɜ ɋɌɈɬɚɤɨɟɠɟɤɚɤ ɢ ɜ ɤɥɚɫɫɢɱɟɫɤɨɣ ɦɟɯɚɧɢɤɟ

Ɂɚɤɨɧ ɫɥɨɠɟɧɢɹ ɫɤɨɪɨɫɬɟɣ ɜ ɋɌɈ

Ɋɚɫɫɦɨɬɪɢɦ ɞɜɢɠɟɧɢɟ ɧɟɤɨɬɨɪɨɣ ɱɚɫɬɢɰɵ ɜ ɫɢɫɬɟɦɚɯ ɨɬɫɱɟɬɚ K ɢK′ Ʉɨɦɩɨɧɟɧɬɵ ɫɤɨɪɨɫɬɢ ɱɚɫɬɢɰɵ ɜ ɷɬɢɯ ɫɢɫɬɟɦɚɯ ɨɬɫɱɟɬɚ

u′

dx′

u′

dy′

u′

dz′

dt′

ɂɡ ɮɨɪɦɭɥ ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ Ʌɨɪɟɧɰɚ ɧɚɣɞɟɦ ɮɨɪɦɭɥɵ ɞɥɹ ɞɢɮɮɟɪɟɧɰɢɚɥɨɜ


	dz − V dt		dt −	V dz
dx′ = dx dy′ = dy dz′ =		dt′ =		c

	− V		− V

	c			c

ɉɨɞɫɬɚɜɥɹɹ ɞɚɧɧɵɟ ɮɨɪɦɭɥɵ ɜ ɮɨɪɦɭɥɵ ɧɚɣɞɟɦ

′

−

′

−

u′

+ V

ux =

uy =

uz =

′

V uz

Ⱦɚɧɧɵɟ ɮɨɪɦɭɥɵ ɧɚɡɵɜɚɸɬɫɹɪɟɥɹɬɢɜɢɫɬɫɤɢɦ ɡɚɤɨɧɨɦ ɫɥɨɠɟɧɢɹ ɫɤɨɪɨ ɫɬɟɣȼ ɱɚɫɬɧɨɦ ɫɥɭɱɚɟ ɦɚɥɵɯ ɫɤɨɪɨɫɬɟɣ(v << c)ɨɧɢ ɩɟɪɟɯɨɞɹɬ ɜ ɤɥɚɫɫɢɱɟɫɤɢɣ ɡɚɤɨɧ ɫɥɨɠɟɧɢɹ ɫɤɨɪɨɫɬɟɣ

76

Лекция13.ПОСТУЛАТЫСТОИИХСЛЕДСТВИЯ

ɉɭɫɬɶ ɱɚɫɬɢɰɚ ɜ ɫɢɫɬɟɦɟ K′ ɞɜɢɠɟɬɫɹ ɫɨ ɫɤɨɪɨɫɬɶɸ u′ = u′ = c Ɍɨɝɞɚ ɢɡ

ɮɨɪɦɭɥ ɧɚɯɨɞɢɦ ɟɟ ɫɤɨɪɨɫɬɶ ɜ ɫɢɫɬɟɦɟK

u = uz =		c + V			= c
	+		V c
				c

ɱɬɨ ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɜɬɨɪɨɦɭ ɩɨɫɬɭɥɚɬɭ ɋɌɈ ɂɡ ɮɨɪɦɭɥ ɫɥɟɞɭɟɬ ɱɬɨ ɫɭɦɦɚ ɞɜɭɯ ɫɤɨɪɨɫɬɟɣ ɤɚɠɞɚɹ ɢɡ ɤɨɬɨɪɵɯ ɦɟɧɶɲɟ ɫɤɨɪɨɫɬɢ ɫɜɟɬɚ ɜɫɟɝɞɚ ɦɟɧɶ ɲɟ ɫɤɨɪɨɫɬɢ ɫɜɟɬɚ ɋɭɦɦɚ ɞɜɭɯ ɫɤɨɪɨɫɬɟɣ ɨɞɧɚ ɢɡ ɤɨɬɨɪɵɯ ɪɚɜɧɚ ɫɤɨɪɨɫɬɢ ɫɜɟ ɬɚ ɚ ɞɪɭɝɚɹ ɦɟɧɶɲɟ ɫɤɨɪɨɫɬɢ ɫɜɟɬɚ ɪɚɜɧɚ ɫɤɨɪɨɫɬɢ ɫɜɟɬɚ

ɅȿɄɐɂə ɆȺɌȿɆȺɌɂɑȿɋɄɂɃ ȺɉɉȺɊȺɌ ɋɌɈ

ɂɧɬɟɪɜɚɥ

Ɋɚɫɫɦɨɬɪɢɦ ɞɜɚ ɫɨɛɵɬɢɹ (x y z t ) (x y z t ) ɜ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ K ɢ ɜ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ K′ ɉɪɢ ɩɟɪɟɯɨɞɟ ɨɬ ɫɢɫɬɟɦɵ K ɤ ɫɢɫɬɟɦɟ K′ ɤɨɨɪɞɢɧɚɬɵ ɫɨɛɵɬɢɣ ɩɪɟɨɛɪɚɡɭɸɬɫɹ ɜ ɫɨɝɥɚɫɢɢ ɫ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹɦɢ Ʌɨɪɟɧɰɚ


	z − V t		t −	V		z
x′ = x y′ = y z′ =	z − V t	t′ =	t −	c		z
	− V		− V

	c				c

ɂɧɬɟɪɜɚɥɨɦɦɟɠɞɭ ɫɨɛɵɬɢɹɦɢ ɢ ɧɚɡɵɜɚɟɬɫɹ ɜɟɥɢɱɢɧɚ

S = c (t − t ) − (x − x ) − (y − y ) − (z − z )

Ɉɞɧɢɦ ɢɡ ɫɥɟɞɫɬɜɢɣ ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ Ʌɨɪɟɧɰɚ ɹɜɥɹɟɬɫɹ ɬɨ ɱɬɨ ɨɧɢ ɧɟ ɢɡ

ɦɟɧɹɸɬ ɢɧɬɟɪɜɚɥ ɦɟɠɞɭ ɫɨɛɵɬɢɹɦɢ ɬ ɟ

S = c

− t )

− (x

− x )

− (y

− y )

− (z

− z )

= S

′

ɂɧɬɟɪɜɚɥɟɫɬɶɢɧɜɚɪɢɚɧɬɩɪɟɨɛɪɚɡɨɜɚɧɢɣɅɨɪɟɧɰɚɢ ɹɜɥɹɟɬɫɹɚɧɚɥɨɝɨɦ ɤɜɚɞɪɚɬɚ ɪɚɫɫɬɨɹɧɢɹ ɦɟɠɞɭ ɞɜɭɦɹ ɬɨɱɤɚɦɢ ɩɪɨɫɬɪɚɧɫɬɜɚ ɜ ɨɛɵɱɧɨɣ ɝɟɨɦɟɬɪɢɢ

ɂɧɬɟɪɜɚɥɵ

S >

ɧɚɡɵɜɚɸɬɫɹ ɜɪɟɦɟɧɢɩɨɞɨɛɧɵɦɢ ɂɧɬɟɪɜɚɥɵ

S <

ɧɚɡɵɜɚɸɬɫɹ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɨɩɨɞɨɛɧɵɦɢ ȿɫɥɢ

S =

ɢɧɬɟɪɜɚɥ ɧɚɡɵɜɚɟɬɫɹ ɫɜɟɬɨɩɨɞɨɛɧɵɦ ɢɡɨɬɪɨɩɧɵɦ

ȿɫɥɢ ɞɜɚ ɫɨɛɵɬɢɹ ɪɚɡɞɟɥɟɧɵ ɜɪɟɦɟɧɢɩɨɞɨɛɧɵɦ ɢɧɬɟɪɜɚɥɨɦ ɬɨ ɨɧɢ ɦɨɝɭɬ ɧɚɯɨɞɢɬɶɫɹ ɜ ɩɪɢɱɢɧɧɨ ɫɥɟɞɫɬɜɟɧɧɨɣ ɫɜɹɡɢ ɞɪɭɝ ɫ ɞɪɭɝɨɦ ɇɚɩɪɢɦɟɪ ɦɨɝɭɬ ɩɪɨɢɫɯɨɞɢɬɶ ɫ ɨɞɧɨɣ ɢ ɬɨɣ ɠɟ ɞɜɢɠɭɳɟɣɫɹ ɱɚɫɬɢɰɟɣ Ⱦɥɹ ɫɨɛɵɬɢɣ ɪɚɡɞɟɥɟɧ ɧɵɯ ɬɚɤɢɦ ɢɧɬɟɪɜɚɥɨɦ ɦɨɠɧɨ ɜɵɛɪɚɬɶ ɂɋɈ ɜ ɤɨɬɨɪɨɣ ɨɧɢ ɹɜɥɹɸɬɫɹ ɨɞɧɨɦɟ ɫɬɧɵɦɢ

ȿɫɥɢɞɜɚɫɨɛɵɬɢɹɪɚɡɞɟɥɟɧɵɩɪɨɫɬɪɚɧɫɬɜɟɧɧɨɩɨɞɨɛɧɵɦɢɧɬɟɪɜɚɥɨɦ ɬɨ ɨɧɢ ɧɟ ɦɨɝɭɬ ɧɚɯɨɞɢɬɶɫɹ ɜ ɩɪɢɱɢɧɧɨ ɫɥɟɞɫɬɜɟɧɧɨɣ ɫɜɹɡɢ ɞɪɭɝ ɫ ɞɪɭɝɨɦ Ⱦɥɹ ɫɨɛɵɬɢɣ ɪɚɡɞɟɥɟɧɧɵɯ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɨɩɨɞɨɛɧɵɦ ɢɧɬɟɪɜɚɥɨɦ ɧɟɥɶɡɹ ɜɵɛɪɚɬɶ ɂɋɈ ɜ ɤɨɬɨɪɨɣ ɨɧɢ ɹɜɥɹɸɬɫɹ ɨɞɧɨɦɟɫɬɧɵɦɢ

ɋɜɨɣɫɬɜɨ ɢɧɜɚɪɢɚɧɬɧɨɫɬɢ ɢɧɬɟɪɜɚɥɚ ɫɜɹɡɵɜɚɟɬ ɜɨɟɞɢɧɨ ɩɪɨɫɬɪɚɧɫɬ

ɜɟɧɧɵɟ ɢ ɜɪɟɦɟɧɧɭɸ ɤɨɨɪɞɢɧɚɬɵ ɫɨɛɵɬɢɣ Ɉɧɢ ɫɜɹɡɚɧɵ ɞɪɭɝ ɫ ɞɪɭɝɨɦ ɫɨɨɬɧɨ ɲɟɧɢɟɦ ɤɨɬɨɪɨɟ ɹɜɥɹɟɬɫɹ ɢɧɜɚɪɢɚɧɬɨɦ ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ Ʌɨɪɟɧɰɚ ȼ ɧɟɪɟɥɹɬɢɜɢɫɬɫɤɨɦ ɫɥɭɱɚɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ Ʌɨɪɟɧɰɚ ɩɟɪɟɯɨɞɹɬ ɜ ɩɪɟ ɨɛɪɚɡɨɜɚɧɢɹ Ƚɚɥɢɥɟɹ ɢ ɩɪɨɫɬɪɚɧɫɬɜɨ ɢ ɜɪɟɦɹ ɧɟ ɫɜɹɡɚɧɵ ɞɪɭɝ ɫ ɞɪɭɝɨɦ ɜ ɧɟɪɟɥɹɬɢɜɢɫɬɫɤɨɦ ɫɥɭɱɚɟ ɢɧɬɟɪɜɚɥ ɪɚɫɩɚɞɚɟɬɫɹ ɧɚ ɞɜɚ ɢɧɜɚɪɢɚɧɬɚ ± ɩɪɨɫɬɪɚɧ ɫɬɜɟɧɧɵɣ ɞɥɢɧɚ ɢ ɜɪɟɦɟɧɧɨɣ ɩɪɨɦɟɠɭɬɨɤ ɜɪɟɦɟɧɢ

78

Лекция14.МАТЕМАТИЧЕСКИЙАППАРАТСТО

ɜɟɤɬɨɪɵ

ȼ ɫɨɝɥɚɫɢɢ ɫ ɩɟɪɜɵɦ ɩɨɫɬɭɥɚɬɨɦ ɋɌɈ ɮɨɪɦɚ ɮɢɡɢɱɟɫɤɨɝɨ ɡɚɤɨɧɚ ɧɟ ɞɨɥɠɧɚ ɦɟɧɹɬɶɫɹ ɩɪɢ ɩɟɪɟɯɨɞɟ ɨɬ ɨɞɧɨɣ ɂɋɈ ɤ ɞɪɭɝɨɣ ɬ ɟ ɮɨɪɦɚ ɡɚɤɨɧɚ ɞɨɥɠɧɚ ɛɵɬɶ ɪɟɥɹɬɢɜɢɫɬɫɤɢ ɤɨɜɚɪɢɚɧɬɧɨɣ Ⱦɥɹ ɡɚɩɢɫɢ ɡɚɤɨɧɨɜ ɦɟɯɚɧɢɤɢ ɜ ɬɚɤɨɣ ɮɨɪɦɟ ɫɥɟɞɭɟɬ ɜɜɟɫɬɢ ɜ ɪɚɫɫɦɨɬɪɟɧɢɟ ɜɟɤɬɨɪɵ ɢ ɫɜɹɡɚɧɧɵɣ ɫ ɧɢɦɢ ɦɚɬɟɦɚɬɢɱɟɫɤɢɣ ɚɩɩɚɪɚɬ

ɪɚɞɢɭɫ ɜɟɤɬɨɪɨɦɧɚɡɵɜɚɟɬɫɹ ɜɟɥɢɱɢɧɚ

X = (x y z ict)= (x x x x )

ɝɞɟ i ± ɦɧɢɦɚɹ ɟɞɢɧɢɰɚ Ʉɨɦɩɨɧɟɧɬɵ ɪɚɞɢɭɫ ɜɟɤɬɨɪɚ ɩɪɟɨɛɪɚɡɭɸɬɫɹ ɜ ɫɨɝɥɚɫɢɢ ɫ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹɦɢ Ʌɨɪɟɧɰɚ

§iV

+ ¨

¸x

x − ¨

¸x

x′ = x x′ = x

x′ =

− V

ȼ ɦɚɬɪɢɱɧɨɣ ɮɨɪɦɟ ɡɚɩɢɫɢ

′

= Λ μν xν

xμ

ɝɞɟ

Λ =

i β γ

− i β γ

β =

ɟɫɬɶ ɦɚɬɪɢɰɚ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ ɢ

− V

ɋɤɚɥɹɪɧɨɟ ɩɪɨɢɡɜɟɞɟɧɢɟ ɜɜɨɞɢɬɫɹ ɩɨ ɨɛɵɱɧɨɦɭ ɩɪɚɜɢɥɭ

(X Y )= x y + x

y + x y + x

= xμ yμ

Ʉɜɚɞɪɚɬ ɪɚɞɢɭɫ ɜɟɤɬɨɪɚ ɪɚɜɟɧ ɢɧɬɟɪɜɚɥɭ ɫɨɛɵɬɢɹ ɫɨ ɡɧɚɤɨɦ ɦɢɧɭɫ

X = −S (X )= −c t + (x + y + z )= x + x + x + x = xμ xμ

Ʉɨɦɩɨɧɟɧɬɵ ɪɚɞɢɭɫ ɜɟɤɬɨɪɚ ɪɚɫɫɦɚɬɪɢɜɚɸɬ ɤɚɤ ɤɨɨɪɞɢɧɚɬɵ ɫɨɛɵɬɢɹ ɜ ɱɟɬɵɪɟɯɦɟɪɧɨɦ ɩɪɨɫɬɪɚɧɫɬɜɟ ɜɪɟɦɟɧɢ ɩɪɨɫɬɪɚɧɫɬɜɨ Ɇɢɧɤɨɜɫɤɨɝɨ Ʉɨɨɪɞɢ ɧɚɬɵ ɱɟɬɵɪɟɯɦɟɪɧɨɝɨ ɩɪɨɫɬɪɚɧɫɬɜɚ ɹɜɥɹɸɬɫɹ ɨɪɬɨɝɨɧɚɥɶɧɵɦɢ ɚ ɮɨɪɦɭɥɚ ɹɜɥɹɟɬɫɹ ɚɧɚɥɨɝɨɦ ɬɟɨɪɟɦɵ ɉɢɮɚɝɨɪɚ ɜ ɨɛɵɱɧɨɣ ɝɟɨɦɟɬɪɢɢ

ȼɜɟɞɟɦ ɩɨɧɹɬɢɟɬɟɧɡɨɪɚɪɚɧɝɚ n ɬɟɧɡɨɪɨɦ ɪɚɧɝɚ n ɧɚɡɵɜɚɟɬɫɹ ɫɨɜɨ ɤɭɩɧɨɫɬɶ n ɱɢɫɟɥ ɤɨɦɩɨɧɟɧɬ ɬɟɧɡɨɪɚ ɤɨɬɨɪɵɟ ɩɪɟɨɛɪɚɡɭɸɬɫɹ ɩɨ ɨɩɪɟɞɟɥɟɧ ɧɵɦ ɩɪɚɜɢɥɚɦ ɜ ɫɨɝɥɚɫɢɢ ɫ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹɦɢ

ɬɟɧɡɨɪɧɭɥɟɜɨɝɨɪɚɧɝɚɧɚɡɵɜɚɟɬɫɹɫɤɚɥɹɪɨɦɋɤɚɥɹɪɧɟɦɟɧɹɟɬɫɹɩɪɢ ɢɡɦɟɧɟɧɢɢ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ ȿɝɨ ɩɪɢɦɟɪɨɦ ɹɜɥɹɟɬɫɹ ɢɧɬɟɪɜɚɥ

М. А. Михайлов. ЛЕКЦИИ ПО КЛАССИЧЕСКОЙ МЕХАНИКЕ

ɬɟɧɡɨɪ ɩɟɪɜɨɝɨ ɪɚɧɝɚ ɟɫɬɶ ɜɟɤɬɨɪ ȿɝɨ ɤɨɦɩɨɧɟɧɬɵ ɩɪɟɨɛɪɚɡɭɸɬɫɹ ɩɨ

ɮɨɪɦɭɥɚɦ ɉɪɢɦɟɪ ɜɟɤɬɨɪɚ ± ɪɚɞɢɭɫ ɜɟɤɬɨɪ Ʉɜɚɞɪɚɬ ɜɟɤɬɨɪɚ ɹɜɥɹ

ɟɬɫɹ ɫɤɚɥɹɪɨɦ ɫɤɚɥɹɪɧɨɟ ɩɪɨɢɡɜɟɞɟɧɢɟ ɞɜɭɯ ɜɟɤɬɨɪɨɜ ɟɫɬɶ ɫɤɚɥɹɪ Ɍɟɧɡɨɪ ɜɬɨɪɨɝɨ ɪɚɧɝɚ ɫɨɞɟɪɠɢɬ ɤɨɦɩɨɧɟɧɬ ȿɝɨ ɩɪɢɦɟɪɨɦ ɹɜɥɹɟɬɫɹ

ɬɟɧɡɨɪ ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɨɝɨ ɩɨɥɹ ɤɨɬɨɪɵɣ ɛɭɞɟɬ ɪɚɫɫɦɨɬɪɟɧ ɜ ɷɥɟɤɬɪɨɞɢɧɚɦɢɤɟ

Ɉɬɦɟɬɢɦ ɱɬɨ ɤɨɦɩɨɧɟɧɬɵ ɬɟɧɡɨɪɚ ɜɬɨɪɨɝɨ ɪɚɧɝɚ ɩɪɟɨɛɪɚɡɭɸɬɫɹ ɩɨ ɡɚɤɨɧɭ

′	= Λ μν Λνη tλη
tμν	= Λ μν Λνη tλη

ɉɪɢɧɰɢɩɨɬɧɨɫɢɬɟɥɶɧɨɫɬɢɩɟɪɜɵɣɩɨɫɬɭɥɚɬɋɌɈɦɨɠɧɨɫɮɨɪɦɭɥɢɪɨ ɜɚɬɶ ɬɟɩɟɪɶ ɜ ɫɥɟɞɭɸɳɟɣ ɮɨɪɦɟ ɮɢɡɢɱɟɫɤɢɟ ɡɚɤɨɧɵ ɞɨɥɠɧɵ ɫɨɞɟɪɠɚɬɶ ɫɥɚɝɚɟ ɦɵɟ ɤɨɬɨɪɵɟ ɩɪɟɨɛɪɚɡɭɸɬɫɹ ɩɪɢ ɢɡɦɟɧɟɧɢɢ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ ɨɞɧɨɬɢɩɧɵɦ ɨɛ ɪɚɡɨɦ ɬ ɟ ɞɨɥɠɧɵ ɛɵɬɶ ɬɟɧɡɨɪɚɦɢ ɨɞɧɨɝɨ ɢ ɬɨɝɨ ɠɟ ɪɚɧɝɚ Ɏɨɪɦɚ ɮɢɡɢɱɟɫɤɨɝɨ ɡɚɤɨɧɚ ɩɪɢ ɷɬɨɦ ɧɟ ɦɟɧɹɟɬɫɹ ȼ ɱɚɫɬɧɨɫɬɢ ɡɚɤɨɧ ɦɨɠɟɬ ɢɦɟɬɶ ɜɢɞ

a = b

ɝɞɟa, b± ɫɤɚɥɹɪɵ ɢɥɢ

aμ = bμ

ɝɞɟaμ bμ ± ɤɨɦɩɨɧɟɧɬɵ ɜɟɤɬɨɪɨɜA ɢB

ɫɤɨɪɨɫɬɶ ɢ ɭɫɤɨɪɟɧɢɟ

ɋɤɨɪɨɫɬɶ ɱɚɫɬɢɰɵ ɨɩɪɟɞɟɥɹɟɬɫɹ ɤɚɤ ɩɪɨɢɡɜɨɞɧɚɹ ɩɨ ɜɪɟɦɟɧɢ ɨɬ ɪɚɞɢɭɫ ɜɟɤɬɨɪɚ ɱɚɫɬɢɰɵ ȼɜɟɞɟɦ ɜɟɤɬɨɪ ɫɤɨɪɨɫɬɢ ɤɚɤ ɩɪɨɢɡɜɨɞɧɭɸ ɨɬ ɪɚɞɢɭɫ ɜɟɤɬɨɪɚ ɩɨ ɧɟɤɨɬɨɪɨɦɭ ɢɧɜɚɪɢɚɧɬɭ ± ɫɤɚɥɹɪɭ ȼɵɛɨɪ ɞɚɧɧɨɝɨ ɢɧɜɚɪɢɚɧɬɚ ɞɨɥ ɠɟɧ ɛɵɬɶ ɨɩɪɟɞɟɥɟɧ ɢɡ ɭɫɥɨɜɢɹ ɱɬɨɛɵ ɬɪɢ ɩɟɪɜɵɟ ɤɨɦɩɨɧɟɧɬɵ ɫɤɨɪɨɫɬɢ ɩɪɢ ɦɚɥɵɯ ɫɤɨɪɨɫɬɹɯv << c ɩɟɪɟɯɨɞɢɥɢ ɜ ɤɨɦɩɨɧɟɧɬɵ ɨɛɵɱɧɨɣ ɫɤɨɪɨɫɬɢ Ⱦɥɹ ɷɬɨɝɨ ɫɥɟɞɭɟɬ ɨɩɪɟɞɟɥɢɬɶ ɫɤɨɪɨɫɬɶ ɤɚɤ ɩɪɨɢɡɜɨɞɧɭɸ ɪɚɞɢɭɫ ɜɟɤɬɨɪɚ ɩɨ ɫɨɛɫɬɜɟɧ ɧɨɦɭ ɜɪɟɦɟɧɢ

U =

ɝɞɟdt

= −

ɂɫɩɨɥɶɡɭɹ ɨɩɪɟɞɟɥɟɧɢɟ ɧɚɣɞɟɦ

i c

U = ¨¨

¸¸

¨¨ −

−

¸¸

Ʉɜɚɞɪɚɬ ɫɤɨɪɨɫɬɢ

= −c

80

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