Лекции по классической механике. Учебное пособие

Министерство образования и науки Российской Федерации Федеральное государственное бюджетное образовательное учреждение

высшего образования

«Московский педагогический государственный университет»

ЛЕКЦИИ ПО КЛАССИЧЕСКОЙ

МЕХАНИКЕ

Учебное пособие

2-е издание, стереотипное электронное

МПГУ Москва • 2024

УДК 53

ББК 22.3 М69

Рецензенты:

И. А. Васильева, доктор физико-математических наук, профессор кафедры общей и экспериментальной физики МПГУ

А. А. Горшелев, кандидат физико-математических наук, старший научный сотрудник ИСАН

Михайлов, Михаил Анатольевич.

М69 Лекции по классической механике: Учебное пособие / М. А. Михайлов. – 2-е изд., стер. электрон. – Москва: МПГУ, 2024. – 92 с. –

Текст: электронный.

ISBN 978-5-4263-0225-9

Учебное пособие содержит лекции по разделу «Классическая механика» (изучается на протяжении двух семестров), включает в себя вопросы ньютоновой механики, аналитической механики и специальной теории относительности. Рассматривается ряд конкретных задач на применение теоретических знаний (рассеяние частиц, законы сохранения, принципы симметрии и др.).

УДК 53

ББК 22.3

ɋɈȾȿɊɀȺɇɂȿ

Ʌɟɤɰɢɹ ȼɜɟɞɟɧɢɟ ɜ ɤɥɚɫɫɢɱɟɫɤɭɸ ɦɟɯɚɧɢɤɭ

Ɉɫɧɨɜɧɵɟ ɡɚɤɨɧɵ ɢ ɬɟɨɪɟɦɵ ɞɢɧɚɦɢɤɢ ɇɶɸɬɨɧɚ

Ʌɟɤɰɢɹ Ɉɫɧɨɜɧɵɟ ɩɨɫɬɭɥɚɬɵ ɞɢɧɚɦɢɤɢ ɇɶɸɬɨɧɚ

Ʌɟɤɰɢɹ Ɉɫɧɨɜɧɵɟ ɬɟɨɪɟɦɵ ɞɢɧɚɦɢɤɢ ɱɚɫɬɢɰɵ

Ʌɟɤɰɢɹ Ɂɚɤɨɧɵ ɢ ɬɟɨɪɟɦɵ ɞɢɧɚɦɢɤɢ ɫɢɫɬɟɦɵ ɱɚɫɬɢɰ

ɇɟɤɨɬɨɪɵɟ ɡɚɞɚɱɢ ɞɢɧɚɦɢɤɢ ɇɶɸɬɨɧɚ

Ⱦɜɢɠɟɧɢɟ ɜ ɰɟɧɬɪɚɥɶɧɨ ɫɢɦɦɟɬɪɢɱɧɨɦ ɩɨɥɟ

Ʌɟɤɰɢɹ Ɉɞɧɨɦɟɪɧɨɟ ɞɜɢɠɟɧɢɟ Ɂɚɞɚɱɚ ɞɜɭɯ ɬɟɥ Ⱦɜɢɠɟɧɢɟ ɱɚɫɬɢɰɵ

ɜ ɰɟɧɬɪɚɥɶɧɨ ɫɢɦɦɟɬɪɢɱɧɨɦ ɩɨɥɟ

Ʌɟɤɰɢɹ ɋɬɨɥɤɧɨɜɟɧɢɹ ɱɚɫɬɢɰ

Ʌɟɤɰɢɹ Ɂɚɞɚɱɚ Ʉɟɩɥɟɪɚ Ɋɚɫɫɟɹɧɢɟ ɱɚɫɬɢɰ

Ɉɫɧɨɜɵ ɚɧɚɥɢɬɢɱɟɫɤɨɣ ɦɟɯɚɧɢɤɢ

Ʌɟɤɰɢɹ Ɉɫɧɨɜɧɵɟ ɩɨɧɹɬɢɹ ɢ ɬɟɨɪɟɦɵ ɚɧɚɥɢɬɢɱɟɫɤɨɣ ɦɟɯɚɧɢɤɢ

Ʌɟɤɰɢɹ Ɉɫɧɨɜɵ ɦɟɯɚɧɢɤɢ Ʌɚɝɪɚɧɠɚ

Ʌɟɤɰɢɹ Ɉɫɧɨɜɵ ɦɟɯɚɧɢɤɢ Ƚɚɦɢɥɶɬɨɧɚ

Ʌɟɤɰɢɹ Ƚɚɪɦɨɧɢɱɟɫɤɢɟ ɤɨɥɟɛɚɧɢɹ ɦɟɯɚɧɢɱɟɫɤɢɯ ɫɢɫɬɟɦ

ɫ ɨɞɧɨɣ ɫɬɟɩɟɧɶɸ ɫɜɨɛɨɞɵ

Ʌɟɤɰɢɹ Ʉɨɥɟɛɚɧɢɹ ɫɢɫɬɟɦ ɫ ɧɟɫɤɨɥɶɤɢɦɢ ɫɬɟɩɟɧɹɦɢ ɫɜɨɛɨɞɵ

Ɉɫɧɨɜɵ ɋɌɈ

Ʌɟɤɰɢɹ ɉɨɫɬɭɥɚɬɵ ɋɌɈ ɢ ɢɯ ɫɥɟɞɫɬɜɢɹ

Ʌɟɤɰɢɹ Ɇɚɬɟɦɚɬɢɱɟɫɤɢɣ ɚɩɩɚɪɚɬ ɋɌɈ

Ʌɟɤɰɢɹ Ⱦɢɧɚɦɢɤɚ ɱɚɫɬɢɰɵ ɜ ɋɌɈ

Ʌɟɤɰɢɹ ɋɢɫɬɟɦɵ ɱɚɫɬɢɰ ɜ ɋɌɈ

3

ɅȿɄɐɂə ȼȼȿȾȿɇɂȿ ȼ ɄɅȺɋɋɂɑȿɋɄɍɘ ɆȿɏȺɇɂɄɍ

Ʉɥɚɫɫɢɱɟɫɤɚɹ ɦɟɯɚɧɢɤɚ ɢɡɭɱɚɟɬ ɦɟɯɚɧɢɱɟɫɤɨɟ ɞɜɢɠɟɧɢɟ ɦɚɤɪɨɫɤɨɩɢɱɟ ɫɤɢɯ ɨɛɴɟɤɬɨɜ ɤɨɬɨɪɵɟ ɞɜɢɠɭɬɫɹ ɫɨ ɫɤɨɪɨɫɬɹɦɢ ɦɧɨɝɨ ɦɟɧɶɲɟ ɫɤɨɪɨɫɬɢ ɫɜɟɬɚ v << c c ā ɦ ɫ ɉɨɞ ɦɚɤɪɨɫɤɨɩɢɱɟɫɤɢɦɢ ɨɛɴɟɤɬɚɦɢ ɩɨɧɢɦɚɸɬɫɹ ɨɛɴɟɤɬɵ ɪɚɡɦɟɪɵ ɤɨɬɨɪɵɯR ≥ − ɦ ɫɩɪɚɜɚ ɨɛɨɡɧɚɱɟɧ ɪɚɡɦɟɪ ɬɢɩɢɱɧɨɣ ɦɨɥɟɤɭɥɵ

Ɏɢɡɢɱɟɫɤɢɟ ɬɟɨɪɢɢ ɢɡɭɱɚɸɳɢɟ ɫɢɫɬɟɦɵ ɬɟɥ ɞɜɢɠɟɧɢɟ ɤɨɬɨɪɵɯ ɩɪɨɢɫ

ɯɨɞɢɬ ɫɨ ɫɤɨɪɨɫɬɹɦɢ ɦɧɨɝɨ ɦɟɧɶɲɢɦɢ ɫɤɨɪɨɫɬɢ ɫɜɟɬɚ ɨɬɧɨɫɹɬɫɹ ɤ ɱɢɫɥɭ ɧɟ

ɪɟɥɹɬɢɜɢɫɬɫɤɢɯ ɬɟɨɪɢɣ ȿɫɥɢ ɫɤɨɪɨɫɬɢ ɱɚɫɬɢɰ ɫɢɫɬɟɦɵ ɫɪɚɜɧɢɦɵ ɫɨ ɫɤɨɪɨ ɫɬɶɸ ɫɜɟɬɚ v ≤ c ɬɨ ɬɚɤɢɟ ɫɢɫɬɟɦɵ ɨɬɧɨɫɹɬɫɹ ɤ ɪɟɥɹɬɢɜɢɫɬɫɤɢɦ ɫɢɫɬɟɦɚɦ ɢ ɨɧɢ ɞɨɥɠɧɵ ɨɩɢɫɵɜɚɬɶɫɹ ɧɚ ɨɫɧɨɜɟ ɪɟɥɹɬɢɜɢɫɬɫɤɢɯ ɬɟɨɪɢɣ Ɉɫɧɨɜɨɣ ɜɫɟɯ ɪɟɥɹɬɢɜɢɫɬɫɤɢɯ ɬɟɨɪɢɣ ɹɜɥɹɟɬɫɹ ɫɩɟɰɢɚɥɶɧɚɹ ɬɟɨɪɢɹ ɨɬɧɨɫɢɬɟɥɶɧɨɫɬɢ ɋɌɈ ȿɫɥɢ ɪɚɡɦɟɪɵ ɢɡɭɱɚɟɦɵɯ ɮɢɡɢɱɟɫɤɢɯ ɨɛɴɟɤɬɨɜ ɦɚɥɵR ≤ − ɦ ɬɨ ɬɚɤɢɟ ɫɢɫ ɬɟɦɵ ɨɬɧɨɫɹɬɫɹ ɤ ɤɜɚɧɬɨɜɵɦ ɫɢɫɬɟɦɚɦ ɢ ɢɯ ɬɟɨɪɢɢ ɩɪɢɧɚɞɥɟɠɚɬ ɤ ɱɢɫɥɭ ɤɜɚɧɬɨɜɵɯ ɬɟɨɪɢɣ

Ɍɚɤɢɦ ɨɛɪɚɡɨɦ ɤɥɚɫɫɢɱɟɫɤɭɸ ɦɟɯɚɧɢɤɭ ɫɥɟɞɭɟɬ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɤɚɤ ɧɟ

ɪɟɥɹɬɢɜɢɫɬɫɤɭɸ ɧɟɤɜɚɧɬɨɜɭɸ ɬɟɨɪɢɸ ɞɜɢɠɟɧɢɹ ɱɚɫɬɢɰ

ɋɢɫɬɟɦɵ ɨɬɫɱɟɬɚ ɢ ɩɪɢɧɰɢɩɵ ɢɧɜɚɪɢɚɧɬɧɨɫɬɢ

Ɇɟɯɚɧɢɱɟɫɤɨɟ ɞɜɢɠɟɧɢɟ± ɷɬɨ ɢɡɦɟɧɟɧɢɟ ɩɨɥɨɠɟɧɢɹ ɬɟɥɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɞɪɭɝɢɯ ɬɟɥ ɫ ɬɟɱɟɧɢɟɦ ɜɪɟɦɟɧɢ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ

ɉɪɨɫɬɪɚɧɫɬɜɨ ɜ ɤɥɚɫɫɢɱɟɫɤɨɣ ɦɟɯɚɧɢɤɟ ɫɱɢɬɚɟɬɫɹ ɬɪɟɯɦɟɪɧɵɦ ɞɥɹ ɨɩ

ɪɟɞɟɥɟɧɢɹ ɩɨɥɨɠɟɧɢɹ ɱɚɫɬɢɰɵ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ ɧɟɨɛɯɨɞɢɦɨ ɡɚɞɚɬɶ ɬɪɢ ɤɨɨɪ

ɞɢɧɚɬɵ ɩɨɞɱɢɧɹɸɳɢɦɫɹ ɝɟɨɦɟɬɪɢɢ ȿɜɤɥɢɞɚ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ ɫɩɪɚɜɟɞɥɢɜɚ

ɬɟɨɪɟɦɚ ɉɢɮɚɝɨɪɚ ɢ ɚɛɫɨɥɸɬɧɵɦ ȼɪɟɦɹ ɨɞɧɨɦɟɪɧɨ ɨɞɧɨɧɚɩɪɚɜɥɟɧɨ ɦɟɧɹ ɟɬɫɹ ɨɬ ɩɪɨɲɥɨɝɨ ɤ ɛɭɞɭɳɟɦɭ ɢ ɚɛɫɨɥɸɬɧɨ Ⱥɛɫɨɥɸɬɧɨɫɬɶ ɩɪɨɫɬɪɚɧɫɬɜɚ ɢ ɜɪɟɦɟɧɢ ɨɡɧɚɱɚɟɬ ɱɬɨ ɢɯ ɫɜɨɣɫɬɜɚ ɧɟ ɡɚɜɢɫɹɬ ɨɬ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɢ ɞɜɢɠɟɧɢɹ ɦɚɬɟɪɢɢ ȼ ɤɥɚɫɫɢɱɟɫɤɨɣ ɦɟɯɚɧɢɤɟ ɩɪɢɧɢɦɚɟɬɫɹ ɫɩɪɚɜɟɞɥɢɜɵɦ ɫɥɟɞɭɸɳɟɟ ɭɬɜɟɪɠɞɟɧɢɟ ɩɪɨɫɬɪɚɧɫɬɜɨ ɢ ɜɪɟɦɹ ɧɟ ɫɜɹɡɚɧɵ ɞɪɭɝ ɫ ɞɪɭɝɨɦ ɢ ɦɨɝɭɬ ɪɚɫ ɫɦɚɬɪɢɜɚɬɶɫɹ ɧɟɡɚɜɢɫɢɦɨ ɞɪɭɝ ɨɬ ɞɪɭɝɚ

Ⱦɜɢɠɟɧɢɟ ɨɬɧɨɫɢɬɟɥɶɧɨ ɢ ɫɥɟɞɨɜɚɬɟɥɶɧɨ ɞɥɹ ɟɝɨ ɨɩɢɫɚɧɢɹ ɧɟɨɛɯɨ ɞɢɦɨ ɜɵɛɪɚɬɶɬɟɥɨ ɨɬɫɱɟɬɚɬ ɟ ɬɟɥɨ ɨɬɧɨɫɢɬɟɥɶɧɨ ɤɨɬɨɪɨɝɨ ɪɚɫɫɦɚɬɪɢɜɚɟɬ ɫɹ ɞɜɢɠɟɧɢɟ ɉɨɫɤɨɥɶɤɭ ɞɜɢɠɟɧɢɟ ɩɪɨɢɫɯɨɞɢɬ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ ɢ ɜɨ ɜɪɟɦɟɧɢ ɬɨ ɞɥɹ ɟɝɨ ɨɩɢɫɚɧɢɹ ɫɥɟɞɭɟɬ ɜɵɛɪɚɬɶ ɬɭ ɢɥɢ ɢɧɭɸ ɫɢɫɬɟɦɭ ɤɨɨɪɞɢɧɚɬ ɢ ɱɚɫɵ ɚɪɢɮɦɟɬɢɡɢɪɨɜɚɬɶ ɩɪɨɫɬɪɚɧɫɬɜɨ ɢ ɜɪɟɦɹ ȼ ɫɢɥɭ ɬɪɟɯɦɟɪɧɨɫɬɢ ɩɪɨɫɬɪɚɧɫɬ ɜɚ ɜ ɤɚɠɞɨɣ ɟɝɨ ɬɨɱɤɟ ɫɨɩɨɫɬɚɜɥɹɸɬɫɹ ɬɪɢ ɱɢɫɥɚ ɤɨɨɪɞɢɧɚɬɵ ȼɵɛɨɪ ɬɨɣ ɢɥɢ ɢɧɨɣ ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ ɨɛɵɱɧɨ ɞɢɤɬɭɟɬɫɹ ɭɫɥɨɜɢɟɦ ɢ ɫɢɦɦɟɬɪɢɟɣ ɩɨ ɫɬɚɜɥɟɧɧɨɣ ɡɚɞɚɱɢ ȼ ɬɟɨɪɟɬɢɱɟɫɤɢɯ ɪɚɫɫɭɠɞɟɧɢɹɯ ɦɵ ɨɛɵɱɧɨ ɛɭɞɟɦ ɢɫɩɨɥɶ ɡɨɜɚɬɶ ɩɪɹɦɨɭɝɨɥɶɧɭɸ ɞɟɤɚɪɬɨɜɭ ɫɢɫɬɟɦɭ ɤɨɨɪɞɢɧɚɬ ɪɢɫ

ȼ ɤɥɚɫɫɢɱɟɫɤɨɣ ɦɟɯɚɧɢɤɟ ɞɥɹ ɢɡɦɟɪɟɧɢɹ ɩɪɨɦɟɠɭɬɤɨɜ ɜɪɟɦɟɧɢ ɜ ɫɢɥɭ ɚɛɫɨɥɸɬɧɨɫɬɢ ɜɪɟɦɟɧɢ ɞɨɫɬɚɬɨɱɧɨ ɧɚɥɢɱɢɹ ɨɞɧɢɯ ɱɚɫɨɜ ɩɨɦɟɳɟɧɧɵɯ ɜ ɧɚɱɚɥɟ ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ ɩɨɞɪɨɛɧɨ ɷɬɨɬ ɜɨɩɪɨɫ ɛɭɞɟɬ ɪɚɫɫɦɨɬɪɟɧ ɜ ɬɟɨɪɢɢ ɨɬɧɨɫɢɬɟɥɶɧɨɫɬɢ Ɍɟɥɨ ɨɬɫɱɟɬɚ ɢ ɫɜɹɡɚɧɧɵɟ ɫ ɷɬɢɦ ɬɟɥɨɦ ɱɚɫɵ ɢ ɦɚɫɲɬɚɛɵ ɫɢɫɬɟɦɚ ɤɨɨɪɞɢɧɚɬ ɨɛɪɚɡɭɸɬɫɢɫɬɟɦɭ ɨɬɫɱɟɬɚ

4

Лекция1.ВВЕДЕНИЕВКЛАССИЧЕСКУЮМЕХАНИКУ

y

;

Ɋɢɫ. 1.1.Ⱦɟɤɚɪɬɨɜɵ ɤɨɨɪɞɢɧɚɬɵ ɱɚɫɬɢɰɵ ɢ ɟɟ ɪɚɞɢɭɫ ɜɟɤɬɨɪ

ȼɜɟɞɟɦ ɩɨɧɹɬɢɟ ɡɚɦɤɧɭɬɨɣ ɮɢɡɢɱɟɫɤɨɣ ɫɢɫɬɟɦɵɁɚɦɤɧɭɬɨɣ ɮɢɡɢɱɟɫɤɨɣ ɫɢɫɬɟɦɨɣɧɚɡɵɜɚɟɬɫɹ ɬɚɤɚɹ ɫɢɫɬɟɦɚ ɦɚɬɟɪɢɚɥɶɧɵɯ ɨɛɴɟɤɬɨɜ ɜ ɤɨɬɨɪɨɣ ɜɫɟ ɨɛɴ ɟɤɬɵ ɫɢɫɬɟɦɵ ɜɡɚɢɦɨɞɟɣɫɬɜɭɸɬ ɦɟɠɞɭ ɫɨɛɨɣ ɧɨ ɧɟ ɜɡɚɢɦɨɞɟɣɫɬɜɭɸɬ ɫ ɨɛɴɟɤɬɚɦɢ ɤɨɬɨɪɵɟ ɧɟ ɜɯɨɞɹɬ ɜ ɫɢɫɬɟɦɭ

Ʉɚɤ ɩɨɤɚɡɵɜɚɸɬ ɷɤɫɩɟɪɢɦɟɧɬɵ ɩɨ ɨɬɧɨɲɟɧɢɸ ɤ ɰɟɥɨɦɭ ɪɹɞɭ ɫɢɫɬɟɦ ɨɬ

ɫɱɟɬɚ ɨɤɚɡɵɜɚɸɬɫɹ ɫɩɪɚɜɟɞɥɢɜɵɦɢ ɫɥɟɞɭɸɳɢɟ ɩɪɢɧɰɢɩɵ ɢɧɜɚɪɢɚɧɬɧɨɫɬɢ ɉɪɢɧɰɢɩ ɢɧɜɚɪɢɚɧɬɧɨɫɬɢ ɨɬɧɨɫɢɬɟɥɶɧɨ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɵɯ ɫɞɜɢɝɨɜɩɪɨ

ɫɬɪɚɧɫɬɜɨ ɨɞɧɨɪɨɞɧɨ ɧɚ ɩɪɨɬɟɤɚɧɢɢ ɩɪɨɰɟɫɫɨɜ ɜɧɭɬɪɢ ɡɚɦɤɧɭɬɨɣ ɮɢɡɢɱɟɫɤɨɣ

ɫɢɫɬɟɦɵ ɧɟ ɫɤɚɡɵɜɚɟɬɫɹ ɟɟ ɦɟɫɬɨ ɩɨɥɨɠɟɧɢɹ ɨɬɧɨɫɢɬɟɥɶɧɨ ɬɟɥɚ ɨɬɫɱɟɬɚ ɉɪɢɧɰɢɩ ɢɧɜɚɪɢɚɧɬɧɨɫɬɢ ɨɬɧɨɫɢɬɟɥɶɧɨ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɵɯ ɩɨɜɨɪɨɬɨɜ

ɩɪɨɫɬɪɚɧɫɬɜɨ ɢɡɨɬɪɨɩɧɨ ɧɚ ɩɪɨɬɟɤɚɧɢɢ ɩɪɨɰɟɫɫɨɜ ɜɧɭɬɪɢ ɡɚɦɤɧɭɬɨɣ ɮɢɡɢɱɟ

ɫɤɨɣ ɫɢɫɬɟɦɵ ɧɟ ɫɤɚɡɵɜɚɟɬɫɹ ɟɟ ɨɪɢɟɧɬɚɰɢɹ ɨɬɧɨɫɢɬɟɥɶɧɨ ɬɟɥɚ ɨɬɫɱɟɬɚ ɉɪɢɧɰɢɩ ɢɧɜɚɪɢɚɧɬɧɨɫɬɢ ɨɬɧɨɫɢɬɟɥɶɧɨ ɜɪɟɦɟɧɧɵɯ ɫɞɜɢɝɨɜɜɪɟɦɹ ɨɞɧɨ

ɪɨɞɧɨ ɧɚ ɩɪɨɬɟɤɚɧɢɢ ɩɪɨɰɟɫɫɨɜ ɜɧɭɬɪɢ ɡɚɦɤɧɭɬɨɣ ɮɢɡɢɱɟɫɤɨɣ ɫɢɫɬɟɦɵ ɧɟ ɫɤɚɡɵɜɚɟɬɫɹ ɜɪɟɦɹ ɧɚɱɚɥɚ ɩɪɨɬɟɤɚɧɢɹ ɩɪɨɰɟɫɫɨɜ

ɉɪɢɧɰɢɩ ɢɧɜɚɪɢɚɧɬɧɨɫɬɢ ɨɬɧɨɫɢɬɟɥɶɧɨ ɡɟɪɤɚɥɶɧɵɯ ɨɬɪɚɠɟɧɢɣ ɩɪɨ ɫɬɪɚɧɫɬɜɨɡɟɪɤɚɥɶɧɨ ɫɢɦɦɟɬɪɢɱɧɨɩɪɨɰɟɫɫɵɩɪɨɬɟɤɚɸɳɢɟɜ ɡɚɦɤɧɭɬɵɯɡɟɪ ɤɚɥɶɧɨ ɫɢɦɦɟɬɪɢɱɧɵɯ ɮɢɡɢɱɟɫɤɢɯ ɫɢɫɬɟɦɚɯ ɫɚɦɢ ɹɜɥɹɸɬɫɹ ɡɟɪɤɚɥɶɧɨ ɫɢɦɦɟɬɪɢɱɧɵɦɢ

Ɍɟ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ ɩɨ ɨɬɧɨɲɟɧɢɸ ɤ ɤɨɬɨɪɵɦ ɩɪɨɫɬɪɚɧɫɬɜɨ ɨɞɧɨɪɨɞɧɨ ɢɡɨɬɪɨɩɧɨ ɢ ɡɟɪɤɚɥɶɧɨ ɫɢɦɦɟɬɪɢɱɧɨ ɢ ɜɪɟɦɹ ɨɞɧɨɪɨɞɧɨ ɧɚɡɵɜɚɸɬɫɹɢɧɟɪɰɢɚ ɥɶɧɵɦɢ ɫɢɫɬɟɦɚɦɢ ɨɬɫɱɟɬɚɂɋɈ

ɉɟɪɜɵɣ ɡɚɤɨɧ ɇɶɸɬɨɧɚɭɬɜɟɪɠɞɚɟɬ ɱɬɨ ɂɋɈ ɫɭɳɟɫɬɜɭɸɬ ɋɭɳɟɫɬɜɭɟɬ ɧɟ ɨɞɧɚ ɚ ɛɟɫɤɨɧɟɱɧɨɟ ɦɧɨɠɟɫɬɜɨ ɂɋɈ Ɍɚ ɫɢɫɬɟɦɚ ɨɬɫɱɟɬɚ

ɤɨɬɨɪɚɹ ɞɜɢɠɟɬɫɹ ɨɬɧɨɫɢɬɟɥɶɧɨ ɂɋɈ ɩɪɹɦɨɥɢɧɟɣɧɨ ɢ ɪɚɜɧɨɦɟɪɧɨ ɫɚɦɚ ɛɭɞɟɬ

ɂɋɈ ɉɪɢɧɰɢɩ ɨɬɧɨɫɢɬɟɥɶɧɨɫɬɢ ɭɬɜɟɪɠɞɚɟɬ ɱɬɨ ɧɚ ɩɪɨɬɟɤɚɧɢɟ ɩɪɨɰɟɫɫɨɜ

ɜ ɡɚɦɤɧɭɬɨɣ ɮɢɡɢɱɟɫɤɨɣ ɫɢɫɬɟɦɟ ɧɟ ɫɤɚɡɵɜɚɟɬɫɹ ɟɟ ɩɪɹɦɨɥɢɧɟɣɧɨɟ ɪɚɜɧɨɦɟɪ

ɧɨɟ ɞɜɢɠɟɧɢɟ ɨɬɧɨɫɢɬɟɥɶɧɨ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ ɡɚɤɨɧɵ ɨɩɢɫɵɜɚɸɳɢɟ ɩɪɨɰɟɫɫɵ ɨɞɢɧɚɤɨɜɵɜ ɪɚɡɧɵɯɂɋɈɫɚɦɢɩɪɨɰɟɫɫɵɛɭɞɭɬɨɞɢɧɚɤɨɜɵɟɫɥɢɨɞɢɧɚɤɨɜɵ ɧɚɱɚɥɶɧɵɟ ɭɫɥɨɜɢɹ

5

М. А. Михайлов. ЛЕКЦИИ ПО КЛАССИЧЕСКОЙ МЕХАНИКЕ

Ɉɫɧɨɜɧɵɟ ɦɨɞɟɥɢ ɢ ɪɚɡɞɟɥɵ ɤɥɚɫɫɢɱɟɫɤɨɣ ɦɟɯɚɧɢɤɢ

ȼ ɤɥɚɫɫɢɱɟɫɤɨɣ ɦɟɯɚɧɢɤɟ ɩɪɢ ɨɩɢɫɚɧɢɢ ɪɟɚɥɶɧɵɯ ɮɢɡɢɱɟɫɤɢɯ ɫɢɫɬɟɦ ɜɜɨɞɢɬɫɹ ɪɹɞ ɚɛɫɬɪɚɤɬɧɵɯ ɩɨɧɹɬɢɣ ɤɨɬɨɪɵɦ ɨɬɜɟɱɚɸɬ ɪɟɚɥɶɧɵɟ ɮɢɡɢɱɟɫɤɢɟ ɨɛɴɟɤɬɵ ȼ ɱɢɫɥɨ ɨɫɧɨɜɧɵɯ ɬɚɤɢɯ ɩɨɧɹɬɢɣ ɜɯɨɞɹɬ ɡɚɦɤɧɭɬɚɹ ɮɢɡɢɱɟɫɤɚɹ ɫɢɫ ɬɟɦɚ ɦɚɬɟɪɢɚɥɶɧɚɹ ɬɨɱɤɚ ɱɚɫɬɢɰɚ ɚɛɫɨɥɸɬɧɨ ɬɜɟɪɞɨɟ ɬɟɥɨ ɫɩɥɨɲɧɚɹ ɫɪɟɞɚ ɢ ɪɹɞ ɞɪɭɝɢɯ

Ɇɚɬɟɪɢɚɥɶɧɚɹ ɬɨɱɤɚ ɱɚɫɬɢɰɚ± ɬɟɥɨ ɪɚɡɦɟɪɚɦɢ ɢ ɜɧɭɬɪɟɧɧɟɣ ɫɬɪɭɤɬɭɪɨɣ ɤɨɬɨɪɨɝɨ ɦɨɠɧɨ ɩɪɟɧɟɛɪɟɱɶ ɩɪɢ ɨɩɢɫɚɧɢɢ ɟɝɨ ɞɜɢɠɟɧɢɹ ɉɪɢ ɷɬɨɦ ɤɚɠɞɚɹ ɱɚɫ ɬɢɰɚ ɯɚɪɚɤɬɟɪɢɡɭɟɬɫɹ ɫɜɨɢɦ ɨɩɪɟɞɟɥɟɧɧɵɦ ɧɚɛɨɪɨɦ ɩɚɪɚɦɟɬɪɨɜ ± ɦɚɫɫɚ ɷɥɟɤ ɬɪɢɱɟɫɤɢɣɡɚɪɹɞȼ ɦɨɞɟɥɢɦɚɬɟɪɢɚɥɶɧɨɣɬɨɱɤɢɧɟɪɚɫɫɦɚɬɪɢɜɚɸɬɫɹɫɬɪɭɤɬɭɪ ɧɵɟɜɧɭɬɪɟɧɧɢɟɯɚɪɚɤɬɟɪɢɫɬɢɤɢɱɚɫɬɢɰɦɨɦɟɧɬɢɧɟɪɰɢɢɞɢɩɨɥɶɧɵɣɦɨɦɟɧɬ ɫɨɛɫɬɜɟɧɧɵɣ ɦɨɦɟɧɬ ɫɩɢɧ ɢ ɞɪ ɉɨɥɨɠɟɧɢɟ ɱɚɫɬɢɰɵ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ ɯɚɪɚɤɬɟɪɢ ɡɭɟɬɫɹ ɬɪɟɦɹ ɱɢɫɥɚɦɢ ɤɨɨɪɞɢɧɚɬɚɦɢ ɢɥɢ ɪɚɞɢɭɫ ɜɟɤɬɨɪɨɦr ɪɢɫ

Ⱥɛɫɨɥɸɬɧɨ ɬɜɟɪɞɨɟ ɬɟɥɨ

• ɫɢɫɬɟɦɚ ɦɚɬɟɪɢɚɥɶɧɵɯ ɬɨɱɟɤ ɪɚɫɫɬɨɹɧɢɹ ɦɟɠɞɭ ɤɨɬɨɪɵɦɢ ɧɟ ɦɟɧɹɸɬɫɹ ɜ ɩɪɨɰɟɫɫɟ ɢɯ ɞɜɢɠɟɧɢɹ

• ɬɟɥɨ ɞɟɮɨɪɦɚɰɢɹɦɢ ɤɨɬɨɪɨɝɨ ɦɨɠɧɨ ɩɪɟɧɟɛɪɟɱɶ Ɋɟɚɥɶɧɵɣ ɮɢɡɢɱɟɫɤɢɣ ɩɪɨɰɟɫɫ ɪɚɫɫɦɚɬɪɢɜɚɟɬɫɹ ɤɚɤ ɧɟɩɪɟɪɵɜɧɚɹ ɩɨɫɥɟ

ɞɨɜɚɬɟɥɶɧɨɫɬɶ ɷɥɟɦɟɧɬɚɪɧɵɯ ɫɨɛɵɬɢɣ ɗɥɟɦɟɧɬɚɪɧɨɟ ɫɨɛɵɬɢɟ± ɷɬɨ ɹɜɥɟɧɢɟ ɫ ɧɭɥɟɜɨɣ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɨɣ ɩɪɨɬɹ

ɠɟɧɧɨɫɬɶɸ ɢ ɧɭɥɟɜɨɣ ɞɥɢɬɟɥɶɧɨɫɬɶɸ ɧɚɩɪɢɦɟɪ ɩɨɩɚɞɚɧɢɟ ɩɭɥɢ ɜ ɦɢɲɟɧɶ ɋɨɛɵɬɢɟ ɯɚɪɚɤɬɟɪɢɡɭɟɬɫɹ ɱɟɬɵɪɶɦɹ ɱɢɫɥɚɦɢ ɤɨɨɪɞɢɧɚɬɚɦɢ ɬɪɢ ɩɪɨɫɬɪɚɧɫɬ ɜɟɧɧɵɟ ɤɨɨɪɞɢɧɚɬɵ ɢɥɢ ɪɚɞɢɭɫ ɜɟɤɬɨɪ ɢ ɨɞɧɚ ɜɪɟɦɟɧɧɚɹ ɤɨɨɪɞɢɧɚɬɚ r& t Ⱦɜɢɠɟɧɢɟɱɚɫɬɢɰɵɩɪɢɷɬɨɦɩɪɟɞɫɬɚɜɥɹɟɬɫɹɤɚɤɧɟɩɪɟɪɵɜɧɚɹɩɨɫɥɟɞɨɜɚɬɟɥɶ ɧɨɫɬɶ ɫɥɟɞɭɸɳɢɯ ɷɥɟɦɟɧɬɚɪɧɵɯ ɫɨɛɵɬɢɣ ɩɪɨɯɨɠɞɟɧɢɟ ɱɚɫɬɢɰɵ ɱɟɪɟɡ ɞɚɧɧɭɸ ɬɨɱɤɭ ɩɪɨɫɬɪɚɧɫɬɜɚ ɜ ɞɚɧɧɨɟ ɜɪɟɦɹ

Ɂɚɤɨɧ ɞɜɢɠɟɧɢɹ ɱɚɫɬɢɰɵ ɫɱɢɬɚɟɬɫɹ ɡɚɞɚɧɧɵɦ ɟɫɥɢ ɢɡɜɟɫɬɧɚ ɡɚɜɢɫɢɦɨɫɬɶ ɪɚɞɢɭɫ ɜɟɤɬɨɪɚ ɱɚɫɬɢɰɵ ɢɥɢ ɬɪɟɯ ɟɟ ɤɨɨɪɞɢɧɚɬ ɨɬ ɜɪɟɦɟɧɢr& = r& t

ȼ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ ɜɢɞɚ ɢɡɭɱɚɟɦɵɯ ɨɛɴɟɤɬɨɜ ɤɥɚɫɫɢɱɟɫɤɭɸ ɦɟɯɚɧɢɤɭ ɩɨɞ

ɪɚɡɞɟɥɹɸɬ ɧɚ ɦɟɯɚɧɢɤɭ ɱɚɫɬɢɰɵ ɢ ɫɢɫɬɟɦɵ ɱɚɫɬɢɰ ɦɟɯɚɧɢɤɭ ɚɛɫɨɥɸɬɧɨ ɬɜɟɪ ɞɨɝɨ ɬɟɥɚ ɦɟɯɚɧɢɤɭ ɫɩɥɨɲɧɵɯ ɫɪɟɞ ɦɟɯɚɧɢɤɚ ɭɩɪɭɝɢɯ ɬɟɥ ɝɢɞɪɨɦɟɯɚɧɢɤɚ ɚɷɪɨɦɟɯɚɧɢɤɚ

ɉɨ ɯɚɪɚɤɬɟɪɭ ɪɟɲɚɟɦɵɯ ɡɚɞɚɱ ɤɥɚɫɫɢɱɟɫɤɭɸ ɦɟɯɚɧɢɤɭ ɩɨɞɪɚɡɞɟɥɹɸɬ ɧɚ ɤɢɧɟɦɚɬɢɤɭ ɞɢɧɚɦɢɤɭ ɢ ɫɬɚɬɢɤɭɄɢɧɟɦɚɬɢɤɚɢɡɭɱɚɟɬ ɦɟɯɚɧɢɱɟɫɤɨɟ ɞɜɢɠɟɧɢɟ ɱɚɫɬɢɰ ɛɟɡ ɭɱɟɬɚ ɩɪɢɱɢɧ ɜɵɡɵɜɚɸɳɢɯ ɢɡɦɟɧɟɧɢɟ ɯɚɪɚɤɬɟɪɚ ɞɜɢɠɟɧɢɹ ɱɚɫɬɢɰ ɫɢɥ Ɂɚɤɨɧ ɞɜɢɠɟɧɢɹ ɱɚɫɬɢɰ ɫɢɫɬɟɦɵ ɫɱɢɬɚɟɬɫɹ ɡɚɞɚɧɧɵɦ ɉɨ ɷɬɨɦɭ ɡɚɤɨɧɭ ɜ ɤɢɧɟɦɚɬɢɤɟ ɨɩɪɟɞɟɥɹɸɬɫɹ ɫɤɨɪɨɫɬɢ ɭɫɤɨɪɟɧɢɹ ɬɪɚɟɤɬɨɪɢɢ ɞɜɢɠɟɧɢɹ ɱɚɫɬɢɰ ɫɢɫɬɟɦɵ Ⱦɢɧɚɦɢɤɚ ɪɚɫɫɦɚɬɪɢɜɚɟɬ ɦɟɯɚɧɢɱɟɫɤɨɟ ɞɜɢɠɟɧɢɟ ɱɚɫɬɢɰ ɫ ɭɱɟɬɨɦ ɩɪɢɱɢɧ ɜɵɡɵɜɚɸɳɢɯ ɢɡɦɟɧɟɧɢɟ ɯɚɪɚɤɬɟɪɚ ɞɜɢɠɟɧɢɹ ɱɚɫɬɢɰ ɋɢɥɵ ɞɟɣɫɬɜɭɸ ɳɢɟ ɦɟɠɞɭ ɱɚɫɬɢɰɚɦɢ ɫɢɫɬɟɦɵ ɢ ɧɚ ɱɚɫɬɢɰɵ ɫɢɫɬɟɦɵ ɫɨ ɫɬɨɪɨɧɵ ɬɟɥ ɧɟ ɜɤɥɸ ɱɟɧɧɵɯ ɜ ɫɢɫɬɟɦɭ ɫɱɢɬɚɸɬɫɹ ɢɡɜɟɫɬɧɵɦɢ ɉɪɢɪɨɞɚ ɫɢɥ ɜ ɤɥɚɫɫɢɱɟɫɤɨɣ ɦɟɯɚ ɧɢɤɟ ɧɟ ɨɛɫɭɠɞɚɟɬɫɹɋɬɚɬɢɤɚɦɨɠɟɬ ɪɚɫɫɦɚɬɪɢɜɚɬɶɫɹ ɤɚɤ ɱɚɫɬɧɵɣ ɫɥɭɱɚɣ ɞɢ ɧɚɦɢɤɢ ɝɞɟ ɢɡɭɱɚɸɬɫɹ ɭɫɥɨɜɢɹ ɦɟɯɚɧɢɱɟɫɤɨɝɨ ɪɚɜɧɨɜɟɫɢɹ ɱɚɫɬɢɰ ɫɢɫɬɟɦɵ

6

Лекция1.ВВЕДЕНИЕВКЛАССИЧЕСКУЮМЕХАНИКУ

ɉɨ ɫɩɨɫɨɛɭ ɨɩɢɫɚɧɢɹ ɫɢɫɬɟɦ ɦɟɯɚɧɢɤɚ ɞɟɥɢɬɫɹ ɧɚ ɧɶɸɬɨɧɨɜɭ ɢ ɚɧɚɥɢɬɢɱɟɫɤɭɸ ɦɟɯɚɧɢɤɭ

ɉɪɟɨɛɪɚɡɨɜɚɧɢɹ ɤɨɨɪɞɢɧɚɬ ɫɨɛɵɬɢɣ

Ɋɚɫɫɦɨɬɪɢɦ ɤɚɤ ɩɪɟɨɛɪɚɡɭɸɬɫɹ ɤɨɨɪɞɢɧɚɬɵ ɫɨɛɵɬɢɣ r& t ɩɪɢ ɩɟɪɟɯɨɞɟ ɨɬ ɨɞɧɨɣ ɂɋɈ ɤ ɞɪɭɝɨɣ

ɉɪɨɫɬɪɚɧɫɬɜɟɧɧɵɣ ɫɞɜɢɝ ȼ ɞɚɧɧɨɦ ɫɥɭɱɚɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ ɜɵɝɥɹɞɹɬ

ɬɚɤ

r&a → r&a = r&a + δr&

ɝɞɟδr&± ɜɟɤɬɨɪ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɨɝɨ ɫɞɜɢɝɚ ɤɨɬɨɪɵɣ ɧɟ ɡɚɜɢɫɢɬ ɨɬ ɧɨɦɟɪɚ ɫɨɛɵ ɬɢɹ ɢɧɞɟɤɫɚ

ȼɪɟɦɟɧɧɨɣ ɫɞɜɢɝ

r&a → r&a = r&a ta → ta = ta + δt

ɝɞɟδt ± ɜɪɟɦɟɧɧɨɣ ɫɞɜɢɝ ɉɪɨɫɬɪɚɧɫɬɜɟɧɧɵɣ ɩɨɜɨɪɨɬ

r&a → r&a = r&a + δr&a = r&a + >δϕ& r&a @ ta → ta = ta

ɝɞɟδϕ& ± ɜɟɤɬɨɪ ɛɟɫɤɨɧɟɱɧɨ ɦɚɥɨɝɨ ɩɨɜɨɪɨɬɚ ɪɢɫ ȼɪɟɦɟɧɧɚɹ ɢɧɜɟɪɫɢɹ ɨɛɪɚɳɟɧɢɟ ɜɪɟɦɟɧɢ

r&a → r&a = r&a ta → ta = −ta

ɉɪɨɫɬɪɚɧɫɬɜɟɧɧɚɹ ɢɧɜɟɪɫɢɹ ɨɬɪɚɠɟɧɢɟ ɜ ɬɨɱɤɟ

r&a → r&a = −r&a ta → ta = ta

ɉɪɟɨɛɪɚɡɨɜɚɧɢɹ Ƚɚɥɢɥɟɹ Ɋɚɫɫɦɚɬɪɢɜɚɟɦ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ ɤɨɨɪɞɢɧɚɬ ɫɨɛɵɬɢɣ ɩɪɢ ɩɟɪɟɯɨɞɟ ɨɬ ɨɞɧɨɣ ɂɋɈ ɤ ɞɪɭɝɨɣ ɤɨɬɨɪɚɹ ɞɜɢɠɟɬɫɹ ɨɬɧɨɫɢɬɟɥɶɧɨ ɩɟɪɜɨɣ ɩɪɹɦɨɥɢɧɟɣɧɨ ɢ ɪɚɜɧɨɦɟɪɧɨ ɫɨ ɫɤɨɪɨɫɬɶɸV ɪɢɫ

r&a → r&a = r&a − V&ta ta → ta = ta

ɝɞɟ ɜɬɨɪɨɟ ɫɨɨɬɧɨɲɟɧɢɟ ɩɨɫɬɭɥɢɪɭɟɬɫɹ ɢ ɜɵɪɚɠɚɟɬ ɫɨɛɨɣ ɚɛɫɨɥɸɬɧɨɫɬɶ ɜɪɟɦɟɧɢ

Ⱦɢɮɮɟɪɟɧɰɢɪɭɹ ɩɨ ɜɪɟɦɟɧɢ ɩɪɚɜɭɸ ɢ ɥɟɜɭɸ ɱɚɫɬɶ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ ɩɪɨ

ɫɬɪɚɧɫɬɜɟɧɧɵɯ ɤɨɨɪɞɢɧɚɬ ɫ ɭɱɟɬɨɦ ɚɛɫɨɥɸɬɧɨɝɨ ɯɚɪɚɤɬɟɪɚ ɜɪɟɦɟɧɢ ɢɫɩɨɥɶɡɭɹ ɨɩɪɟɞɟɥɟɧɢɟɫɤɨɪɨɫɬɢɤɚɤ ɩɪɨɢɡɜɨɞɧɨɣ ɨɬ ɪɚɞɢɭɫ ɜɟɤɬɨɪɚ ɩɨ ɜɪɟɦɟɧɢ ɭɫɥɨɜɢɟ ɱɬɨV FRQVW ɩɨɥɭɱɚɟɦ ɤɥɚɫɫɢɱɟɫɤɢɣ ɡɚɤɨɧ ɫɥɨɠɟɧɢɹ ɫɤɨɪɨɫɬɟɣ

v&a = v&a − V&

Ɂɞɟɫɶ ɫɥɟɞɭɟɬ ɨɫɨɛɨ ɨɛɪɚɬɢɬɶ ɜɧɢɦɚɧɢɟ ɧɚ ɬɨ ɨɛɫɬɨɹɬɟɥɶɫɬɜɨ ɱɬɨ ɩɪɢ ɜɵ ɜɨɞɟ ɩɨɫɥɟɞɧɟɝɨ ɫɨɨɬɧɨɲɟɧɢɹɧɟɨɛɯɨɞɢɦɨɩɪɢɧɢɦɚɬɶ ɜɨ ɜɧɢɦɚɧɢɟ ɩɨɫɬɭɥɚɬ ɨɛ ɚɛɫɨɥɸɬɧɨɦ ɯɚɪɚɤɬɟɪɟ ɜɪɟɦɟɧɢ

7

М. А. Михайлов. ЛЕКЦИИ ПО КЛАССИЧЕСКОЙ МЕХАНИКЕ

ɩɨɜɨɪɨɬɚ

ɩɨɜɨɪɨɬɚ

Ⱦɢɮɮɟɪɟɧɰɢɪɭɹɩɨɜɪɟɦɟɧɢɟɳɟɪɚɡɢɫɩɨɥɶɡɭɹɨɩɪɟɞɟɥɟɧɢɟɭɫɤɨɪɟɧɢɹ ɤɚɤ ɩɪɨɢɡɜɨɞɧɨɝɨ ɨɬ ɫɤɨɪɨɫɬɢ ɩɨ ɜɪɟɦɟɧɢ ɩɨɥɭɱɢɦ ɱɬɨ ɭɫɤɨɪɟɧɢɟ ɨɞɢɧɚɤɨɜɨ ɩɨ ɨɬɧɨɲɟɧɢɸ ɤ ɪɚɡɧɵɦ ɂɋɈ ɢɧɜɚɪɢɚɧɬɧɨ ɨɬɧɨɫɢɬɟɥɶɧɨ ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ Ƚɚ ɥɢɥɟɹ Ⱦɚɧɧɨɟ ɭɬɜɟɪɠɞɟɧɢɟ ɦɚɬɟɦɚɬɢɱɟɫɤɢ ɜɵɪɚɠɚɟɬ ɫɨɛɨɣ ɩɪɢɧɰɢɩ ɨɬɧɨɫɢ ɬɟɥɶɧɨɫɬɢ ɜ ɤɥɚɫɫɢɱɟɫɤɨɣ ɦɟɯɚɧɢɤɟ

ɋ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɬɨɱɤɢ ɡɪɟɧɢɹ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ ± ɨɛɪɚɡɭɸɬ ɝɪɭɩɩɭ Ⱦɟɣɫɬɜɢɬɟɥɶɧɨ ɞɚɧɧɚɹ ɝɪɭɩɩɚ ɫɨɞɟɪɠɢɬ ɜ ɫɟɛɟ ɟɞɢɧɢɱɧɨɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɟ ± ɬɨɠɞɟɫɬɜɟɧɧɨɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɟ ɨɬɜɟɱɚɸɳɟɟ ɨɬɫɭɬɫɬɜɢɸ ɩɟɪɟɯɨɞɚ ɨɬ ɨɞɧɨɣ ɫɢɫɬɟɦɵ ɤ ɞɪɭɝɨɣ ɞɥɹ ɤɚɠɞɨɝɨ ɢɡ ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ ± ɫɭɳɟɫɬɜɭɟɬ ɨɛɪɚɬɧɨɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɟ ɤɨɬɨɪɨɟ ɩɟɪɟɜɨɞɢɬ ɫɢɫɬɟɦɭ ɜ ɢɫɯɨɞɧɨɟ ɫɨɫɬɨɹɧɢɟ Ɉɩɟɪɚɰɢɹ ɭɦɧɨɠɟɧɢɹ ɤɨɦɩɨɡɢɰɢɢ ɜɜɨɞɢɬɫɹ ɤɚɤ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɟ ɩɪɢɦɟɧɟɧɢɟ ɫɨɨɬɜɟɬ ɫɬɜɭɸɳɢɯ ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ ɋɥɟɞɭɟɬ ɨɫɨɛɨ ɨɛɪɚɬɢɬɶ ɜɧɢɦɚɧɢɟ ɧɚ ɬɨ ɱɬɨ ɝɪɭɩɩɚ ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ ɜɪɚɳɟɧɢɹ ɧɟ ɩɨɞɱɢɧɹɟɬɫɹ ɤɨɦɦɭɬɚɬɢɜɧɨɦɭ ɩɟɪɟɫɬɚɧɨɜɨɱɧɨ ɦɭ ɡɚɤɨɧɭ ɬ ɟ ɹɜɥɹɟɬɫɹ ɧɟɚɛɟɥɟɜɨɣ ɉɨɥɧɭɸ ɝɪɭɩɩɭ ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ ± ɧɚ ɡɵɜɚɸɬ ɝɚɥɢɥɟɟɜɨɣ ɝɪɭɩɩɨɣ ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ

ȼɟɤɬɨɪɵ ɢ ɫɤɚɥɹɪɵ

ȼɟɤɬɨɪɨɦ ɧɚɡɵɜɚɟɬɫɹ ɮɢɡɢɱɟɫɤɚɹ ɜɟɥɢɱɢɧɚ ɤɨɬɨɪɚɹ ɩɪɟɨɛɪɚɡɭɟɬɫɹ ɤɚɤ ɪɚɞɢɭɫ ɜɟɤɬɨɪ ɱɚɫɬɢɰɵ ɢ ɯɚɪɚɤɬɟɪɢɡɭɟɬɫɹ ɫɜɨɢɦ ɱɢɫɥɟɧɧɵɦ ɡɧɚɱɟɧɢɟɦ ɢ ɧɚɩɪɚɜɥɟɧɢɟɦɜ ɩɪɨɫɬɪɚɧɫɬɜɟɉɨɨɬɧɨɲɟɧɢɸɤ ɨɩɟɪɚɰɢɢɩɪɨɫɬɪɚɧɫɬɜɟɧɧɨɣ ɢɧɜɟɪɫɢɢ ɜɟɤɬɨɪɵ ɞɟɥɹɬɫɹ ɧɚɢɫɬɢɧɧɵɟɩɨɥɹɪɧɵɟ ɢɩɫɟɜɞɨɜɟɤɬɨɪɵɚɤɫɢɚɥɶ ɧɵɟ ɉɪɢ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɨɣ ɢɧɜɟɪɫɢɢ ɢɫɬɢɧɧɵɣ ɜɟɤɬɨɪ ɦɟɧɹɟɬ ɫɜɨɣ ɡɧɚɤ ɩɫɟɜɞɨɜɟɤɬɨɪ ɧɟ ɢɡɦɟɧɹɟɬɫɹ

ɋɤɚɥɹɪɵɯɚɪɚɤɬɟɪɢɡɭɸɬɫɹ ɬɨɥɶɤɨ ɫɜɨɢɦ ɱɢɫɥɟɧɧɵɦ ɡɧɚɱɟɧɢɟɦ ɉɨ ɨɬɧɨ ɲɟɧɢɸɤ ɨɩɟɪɚɰɢɢɩɪɨɫɬɪɚɧɫɬɜɟɧɧɨɣɢɧɜɟɪɫɢɢɫɤɚɥɹɪɵɞɟɥɹɬɫɹɧɚɢɫɬɢɧɧɵɟ

8

Лекция1.ВВЕДЕНИЕВКЛАССИЧЕСКУЮМЕХАНИКУ

ɢɩɫɟɜɞɨɫɤɚɥɹɪɵɉɪɢ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɨɣ ɢɧɜɟɪɫɢɢ ɢɫɬɢɧɧɵɣ ɫɤɚɥɹɪ ɧɟ ɢɡɦɟɧɹ ɟɬɫɹ ɩɫɟɜɞɨɫɤɚɥɹɪ ɦɟɧɹɟɬ ɫɜɨɣ ɡɧɚɤ

ɉɪɢɦɟɪɵɊɚɞɢɭɫ ɜɟɤɬɨɪ ɫɤɨɪɨɫɬɶ ɭɫɤɨɪɟɧɢɟ ɱɚɫɬɢɰɵ ɹɜɥɹɸɬɫɹ ɢɫɬɢɧ ɧɵɦɢ ɜɟɤɬɨɪɚɦɢ ȼɟɤɬɨɪɵ ɭɝɥɚ ɩɨɜɨɪɨɬɚ ɭɝɥɨɜɨɣ ɫɤɨɪɨɫɬɢ ɭɝɥɨɜɨɝɨ ɭɫɤɨɪɟ ɧɢɹ ± ɩɫɟɜɞɨɜɟɤɬɨɪɵ ȼɟɤɬɨɪɧɨɟ ɩɪɨɢɡɜɟɞɟɧɢɟ ɞɜɭɯ ɢɫɬɢɧɧɵɯ ɜɟɤɬɨɪɨɜ ± ɩɫɟɜ ɞɨɜɟɤɬɨɪ ɜɟɤɬɨɪɧɨɟ ɩɪɨɢɡɜɟɞɟɧɢɟ ɢɫɬɢɧɧɨɝɨ ɜɟɤɬɨɪɚ ɧɚ ɩɫɟɜɞɨɜɟɤɬɨɪ ± ɢɫɬɢɧ ɧɵɣ ɜɟɤɬɨɪ ɋɤɚɥɹɪɧɨɟ ɩɪɨɢɡɜɟɞɟɧɢɟ ɞɜɭɯ ɢɫɬɢɧɧɵɯ ɜɟɤɬɨɪɨɜ ± ɢɫɬɢɧɧɵɣ ɫɤɚ ɥɹɪ ɢɫɬɢɧɧɨɝɨ ɜɟɤɬɨɪɚ ɧɚ ɩɫɟɜɞɨɜɟɤɬɨɪ ± ɩɫɟɜɞɨɫɤɚɥɹɪ

ɋɥɟɞɭɟɬ ɨɬɦɟɬɢɬɶ ɱɬɨ ɜ ɜɟɤɬɨɪɧɨɦ ɢɥɢ ɫɤɚɥɹɪɧɨɦ ɪɚɜɟɧɫɬɜɟ ɫɩɪɚɜɚ ɢ ɫɥɟɜɚɞɨɥɠɧɵɫɬɨɹɬɶɫɥɚɝɚɟɦɵɟɨɞɧɨɣɩɪɢɪɨɞɵɩɨɨɬɧɨɲɟɧɢɸɤ ɨɩɟɪɚɰɢɢ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɨɣ ɢɧɜɟɪɫɢɢ ɢɫɬɢɧɧɵɟ ɫɤɚɥɹɪɵ ɢɥɢ ɩɫɟɜɞɨɫɤɚɥɹɪɵ ɢɫɬɢɧɧɵɟ ɜɟɤɬɨɪɵ ɢɥɢ ɩɫɟɜɞɨɜɟɤɬɨɪɵ

9

ɈɋɇɈȼɇɕȿ ɁȺɄɈɇɕ ɂ ɌȿɈɊȿɆɕ ȾɂɇȺɆɂɄɂ ɇɖɘɌɈɇȺ

ɅȿɄɐɂə ɈɋɇɈȼɇɕȿ ɉɈɋɌɍɅȺɌɕ ȾɂɇȺɆɂɄɂ ɇɖɘɌɈɇȺ

Ⱦɢɧɚɦɢɤɚɢɡɭɱɚɟɬ ɦɟɯɚɧɢɱɟɫɤɨɟ ɞɜɢɠɟɧɢɟ ɱɚɫɬɢɰ ɫ ɭɱɟɬɨɦ ɩɪɢɱɢɧ ɜɵ ɡɵɜɚɸɳɢɯ ɢɡɦɟɧɟɧɢɟ ɯɚɪɚɤɬɟɪɚ ɢɯ ɞɜɢɠɟɧɢɹ ȼ ɨɫɧɨɜɟ ɧɶɸɬɨɧɨɜɨɣ ɦɟɯɚɧɢɤɢ ɥɟɠɚɬ ɧɟɫɤɨɥɶɤɨ ɩɨɫɬɭɥɚɬɨɜ ɤɨɬɨɪɵɟ ɛɵɥɢ ɫɮɨɪɦɭɥɢɪɨɜɚɧɵ ɂ ɇɶɸɬɨɧɨɦ ɢ ɧɨɫɹɬ ɧɚɡɜɚɧɢɟ ɡɚɤɨɧɨɜ ɇɶɸɬɨɧɚ Ɋɚɫɫɦɨɬɪɟɧɢɟ ɞɜɢɠɟɧɢɹ ɱɚɫɬɢɰ ɩɪɨɜɨɞɢɬɫɹ ɩɨ ɨɬɧɨɲɟɧɢɸ ɤ ɂɋɈ ɋɥɭɱɚɢ ɤɨɝɞɚ ɞɜɢɠɟɧɢɹ ɱɚɫɬɢɰ ɪɚɫɫɦɚɬɪɢɜɚɸɬɫɹ ɩɨ ɨɬ ɧɨɲɟɧɢɸ ɤ ɧɟɢɧɟɪɰɢɚɥɶɧɵɦ ɫɢɫɬɟɦɚɦ ɨɬɫɱɟɬɚ ɨɝɨɜɚɪɢɜɚɸɬɫɹ ɨɫɨɛɨ

ɂɧɟɪɰɢɚɥɶɧɵɟ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ ɉɪɢɧɰɢɩ ɨɬɧɨɫɢɬɟɥɶɧɨɫɬɢ

ɂɧɟɪɰɢɚɥɶɧɨɣ ɫɢɫɬɟɦɨɣ ɨɬɫɱɟɬɚ ɧɚɡɵɜɚɟɬɫɹ ɫɢɫɬɟɦɚ ɨɬɫɱɟɬɚ ɩɨ ɨɬɧɨɲɟ ɧɢɸ ɤ ɤɨɬɨɪɨɣ ɩɪɨɫɬɪɚɧɫɬɜɨ ɨɞɧɨɪɨɞɧɨ ɢɡɨɬɪɨɩɧɨ ɡɟɪɤɚɥɶɧɨ ɫɢɦɦɟɬɪɢɱɧɨ ɢ ɜɪɟɦɹ ɨɞɧɨɪɨɞɧɨ ɫɦ ɥɟɤɰɢɸɂɡɨɥɢɪɨɜɚɧɧɚɹ ɱɚɫɬɢɰɚɬ ɟ ɱɚɫɬɢɰɚ ɤɨɬɨ ɪɚɹ ɧɟ ɩɨɞɜɟɪɠɟɧɚ ɤɚɤɢɦ ɥɢɛɨ ɜɨɡɞɟɣɫɬɜɢɹɦ ɫɨ ɫɬɨɪɨɧɵ ɞɪɭɝɢɯ ɨɛɴɟɤɬɨɜ ɜ ɞɚɧɧɨɣ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ ɞɜɢɠɟɬɫɹ ɫ ɩɨɫɬɨɹɧɧɨɣ ɫɤɨɪɨɫɬɶɸ ɬ ɟ ɩɪɹɦɨɥɢɧɟɣɧɨ ɢ ɪɚɜɧɨɦɟɪɧɨ ɢɥɢ ɩɨɤɨɢɬɫɹ Ⱦɚɧɧɨɟ ɭɬɜɟɪɠɞɟɧɢɟ ɧɨɫɢɬ ɧɚɡɜɚɧɢɟɡɚɤɨɧɚ ɢɧɟɪ ɰɢɢ ɢ ɹɜɥɹɟɬɫɹɫɥɟɞɫɬɜɢɟɦɨɞɧɨɪɨɞɧɨɫɬɢɢ ɢɡɨɬɪɨɩɧɨɫɬɢɩɪɨɫɬɪɚɧɫɬɜɚɨɞɧɨ ɪɨɞɧɨɫɬɢ ɜɪɟɦɟɧɢ ɢ ɩɪɢɧɰɢɩɚ ɨɬɧɨɫɢɬɟɥɶɧɨɫɬɢ Ⱦɚɧɧɵɣ ɡɚɤɨɧ ɦɨɠɧɨ ɫɮɨɪɦɭ ɥɢɪɨɜɚɬɶ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ ɭɫɤɨɪɟɧɢɟ ɢɡɨɥɢɪɨɜɚɧɧɨɣ ɱɚɫɬɢɰɵ ɜ ɂɋɈ ɪɚɜɧɨ ɧɭɥɸ

ɉɟɪɜɵɣ ɡɚɤɨɧ ɇɶɸɬɨɧɚɩɨɫɬɭɥɢɪɭɟɬ ɫɭɳɟɫɬɜɨɜɚɧɢɟ ɂɋɈ ɋɭɳɟɫɬɜɨɜɚɧɢɟ ɂɋɈ ɞɨɥɠɧɨ ɩɪɨɜɟɪɹɬɶɫɹ ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɨ ɋɬɪɨɝɨ

ɢɧɟɪɰɢɚɥɶɧɨɣɫɢɫɬɟɦɵɨɬɫɱɟɬɚɧɟɫɭɳɟɫɬɜɭɟɬɌɭɢɥɢɢɧɭɸɫɢɫɬɟɦɭɨɬɫɱɟɬɚ ɦɨɠɧɨ ɫɱɢɬɚɬɶ ɢɧɟɪɰɢɚɥɶɧɨɣ ɥɢɲɶ ɫ ɢɡɜɟɫɬɧɨɣ ɫɬɟɩɟɧɶɸ ɬɨɱɧɨɫɬɢ ɋ ɛɨɥɶɲɨɣ ɫɬɟɩɟɧɶɸ ɬɨɱɧɨɫɬɢ ɢɧɟɪɰɢɚɥɶɧɨɣ ɦɨɠɧɨ ɫɱɢɬɚɬɶ ɫɢɫɬɟɦɭ ɨɬɫɱɟɬɚ ɫɜɹɡɚɧɧɭɸ ɫ Ɂɟɦɥɟɣ ɝɟɨɰɟɧɬɪɢɱɟɫɤɚɹ ɫɢɫɬɟɦɚ ɨɬɫɱɟɬɚ ɋ ɟɳɟ ɛɨɥɶɲɟɣ ɫɬɟɩɟɧɶɸ ɬɨɱɧɨɫɬɢ ɢɧɟɪɰɢɚɥɶɧɨɣ ɦɨɠɧɨ ɫɱɢɬɚɬɶ ɫɢɫɬɟɦɭ ɨɬɫɱɟɬɚ ɫɜɹɡɚɧɧɭɸ ɫ ɋɨɥɧɰɟɦ ɝɟɥɢɨ ɰɟɧɬɪɢɱɟɫɤɚɹ ɫɢɫɬɟɦɚ ɨɬɫɱɟɬɚ

Ɉɩɵɬ ɩɨɤɚɡɵɜɚɟɬ ɱɬɨ ɫɢɫɬɟɦɚ ɨɬɫɱɟɬɚ ɤɨɬɨɪɚɹ ɞɜɢɠɟɬɫɹ ɩɪɹɦɨɥɢɧɟɣɧɨ

ɢ ɪɚɜɧɨɦɟɪɧɨ ɨɬɧɨɫɢɬɟɥɶɧɨ ɂɋɈ ɫɚɦɚ ɛɭɞɟɬ ɢɧɟɪɰɢɚɥɶɧɨɣ ɫ ɬɨɣ ɠɟ ɫɬɟɩɟɧɶɸ

ɬɨɱɧɨɫɬɢ Ɍɚɤɢɦ ɨɛɪɚɡɨɦ ɫɭɳɟɫɬɜɭɟɬ ɰɟɥɨɟ ɦɧɨɠɟɫɬɜɨ ɂɋɈ ɉɪɢɧɰɢɩ ɨɬɧɨɫɢɬɟɥɶɧɨɫɬɢ ɭɬɜɟɪɠɞɚɟɬ ɪɚɜɧɨɩɪɚɜɢɟ ɜɫɟɯ ɂɋɈ ɫɪɟɞɢ

ɂɋɈ ɧɟɬ ɤɚɤɨɣ ɥɢɛɨ ɨɫɨɛɨ ɜɵɞɟɥɟɧɧɨɣ ɩɪɟɞɩɨɱɬɢɬɟɥɶɧɨɣ ɞɥɹ ɨɩɢɫɚɧɢɹ ɪɚɡ

ɥɢɱɧɵɯ ɮɢɡɢɱɟɫɤɢɯ ɩɪɨɰɟɫɫɨɜ ȼɩɟɪɜɵɟ ɩɪɢɧɰɢɩ ɨɬɧɨɫɢɬɟɥɶɧɨɫɬɢ ɛɵɥ ɫɮɨɪ

ɦɭɥɢɪɨɜɚɧ Ƚɚɥɢɥɟɟɦ ɢ ɨɯɜɚɬɵɜɚɥ ɦɧɨɠɟɫɬɜɨ ɦɟɯɚɧɢɱɟɫɤɢɯ ɩɪɨɰɟɫɫɨɜ ɦɟɯɚɧɢ

ɱɟɫɤɢɣ ɩɪɢɧɰɢɩ ɨɬɧɨɫɢɬɟɥɶɧɨɫɬɢ ȼ ;; ɜɟɤɟ Ⱥ ɗɣɧɲɬɟɣɧɨɦ ɩɪɢɧɰɢɩ ɨɬɧɨɫɢ

ɬɟɥɶɧɨɫɬɢ ɛɵɥ ɪɚɫɩɪɨɫɬɪɚɧɟɧ ɧɚ ɜɫɟ ɩɪɨɰɟɫɫɵ ɜ ɩɪɢɪɨɞɟ ɨɩɬɢɱɟɫɤɢɟ ɷɥɟɤɬɪɨ ɦɚɝɧɢɬɧɵɟ ɢ ɞɪ Ⱦɚɧɧɵɣ ɩɪɢɧɰɢɩ ɥɟɠɢɬ ɜ ɨɫɧɨɜɟ ɩɨɫɬɪɨɟɧɢɹ ɫɩɟɰɢɚɥɶɧɨɣ ɬɟɨɪɢɢ ɨɬɧɨɫɢɬɟɥɶɧɨɫɬɢ ɢ ɹɜɥɹɟɬɫɹ ɨɛɨɛɳɟɧɢɟɦ ɨɝɪɨɦɧɨɝɨ ɱɢɫɥɚ ɷɤɫɩɟɪɢɦɟɧ ɬɚɥɶɧɵɯ ɮɚɤɬɨɜ

ɉɪɢɧɰɢɩ ɨɬɧɨɫɢɬɟɥɶɧɨɫɬɢ ɞɨɩɭɫɤɚɟɬ ɫɥɟɞɭɸɳɭɸ ɟɝɨ ɮɨɪɦɭɥɢɪɨɜɤɭ ɡɚ

ɤɨɧɵ ɩɪɢɪɨɞɵ ɜ ɬɨɦ ɱɢɫɥɟ ɦɟɯɚɧɢɤɢ ɞɨɥɠɧɵ ɛɵɬɶ ɤɨɜɚɪɢɚɧɬɧɵ ɩɨ ɨɬɧɨɲɟ