Уравнение Хилла

Уравнение Хилла (Дж.Хилл, 1886[1]) — линейное дифференциальное уравнение второго порядка:

где f(t) периодическая функция. Важными частными случаями уравнения Хилла являются уравнение Матьё и уравнение Мейснера.

Уравнение Хилла можно представить, как уравнение колебательной системы, где собственная частота колебаний меняется по периодическому закону f(t).

Уравнение Хилла очень важно для понимания устойчивости движения в осцилляторных системах. В зависимости от конкретной формы периодической функции f(t) решения могут иметь вид устойчивых квазипериодических колебаний, либо колебания будут раскачиваться с нарастающей экспоненциально амплитудой.

Применение

Уравнение Хилла позволяет также понять расщепление энергетических уровней электронов в периодическом поле кристаллической решётки.

В физике ускорителей уравнение Хилла описывает поперечную линейную динамику частиц в фокусирующих магнитных полях (бетатронные колебания).

В основе теории работы гиперболоидных масс-спектрометров также лежат варианты уравнения Хилла: уравнение Матьё и уравнение Мейснера (в зависимости от формы изменения во времени подаваемых на электроды потенциалов).

См. также

Параметрический осциллятор

Ссылки

  1. "On the Part of the Motion of Lunar Perigee Which is a Function of the Mean Motions of the Sun and Moon", Acta Math. 8: 1–36.

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