Spinning tops and magnetic orbits
- 作者: Novikov S.P.1
-
隶属关系:
- Steklov Mathematical Institute of Russian Academy of Sciences
- 期: 卷 75, 编号 6 (2020)
- 页面: 153-161
- 栏目: Articles
- URL: https://journals.rcsi.science/0042-1316/article/view/133640
- DOI: https://doi.org/10.4213/rm9977
- ID: 133640
如何引用文章
开放存取
##reader.subscriptionAccessGranted##
订阅存取
详细
A number of directions were initiated by the author and his students in their papers of 1981–1982. However, one of them, concerning the properties of closed orbits on the sphere $S^2$ and in the groups $S^3$ and $\operatorname{SO}_3$, has not been sufficiently developed. This paper revives the discussion of these questions, states unsolved problems, and explains what was regarded as fallacies in old papers. In general, magnetic orbits have been poorly discussed in the literature on dynamical systems and theoretical mechanics, but Grinevich has pointed out that in theoretical physics one encounters similar situations in the theory related to particle accelerators such as proton cyclotrons. It is interesting to look at Chap. III of Landau and Lifshitz's Theoretical physics, vol. 2, Field theory (translated into English as The classical theory of fields [12]), where mathematical relatives of our situations occur, but the physics is completely different and there are actual strong magnetic fields.Bibliography: 12 titles.
作者简介
Sergei Novikov
Steklov Mathematical Institute of Russian Academy of Sciences
Email: snovikov@mi-ras.ru
Doctor of physico-mathematical sciences, Professor
参考
- Б. А. Дубровин, С. П. Новиков, А. Т. Фоменко, Современная геометрия. Методы теории гомологий, Наука, М., 1984, 344 с.
- С. П. Новиков, “Многозначные функции и функционалы. Аналог теории Морса”, Докл. АН СССР, 260:1 (1981), 31–35
- С. П. Новиков, “Гамильтонов формализм и многозначный аналог теории Морса”, УМН, 37:5(227) (1982), 3–49
- В. В. Козлов, “Принцип наименьшего действия и периодические решения в задачах классической механики”, ПММ, 40:3 (1976), 399–407
- В. В. Козлов, Д. А. Онищенко, “Неинтегрируемость уравнений Кирхгофа”, Докл. АН СССР, 266:6 (1982), 1298–1300
- М. П. Харламов, “Понижение порядка в механических системах с симметрией”, Мех. тверд. тела, 8 (1976), 4–18
- С. П. Новиков, И. А. Тайманов, “Периодические экстремали многозначных или не всюду положительных функционалов”, Докл. АН СССР, 274:1 (1984), 26–28
- P. G. Grinevich, S. P. Novikov, “Nonselfintersecting magnetic orbits on the plane. Proof of the overthrowing of cycles principle”, Topics in topology and mathematical physics, Amer. Math. Soc. Transl. Ser. 2, 170, Adv. Math. Sci., 27, Amer. Math. Soc., Providence, RI, 1995, 59–82
- В. В. Козлов, “Вариационное исчисление в целом и классическая механика”, УМН, 40:2(242) (1985), 33–60
- В. В. Козлов, “Замкнутые орбиты и хаотическая динамика заряда в периодическом электромагнитном поле”, Regul. Chaotic Dyn., 2:1 (1997), 3–12
- C. C. Conley, E. Zehnder, “The Birkhoff–Lewis fixed point theorem and a conjecture of V. I. Arnold”, Invent. Math., 73:1 (1983), 33–49
- Л. Д. Ландау, Е. М. Лифшиц, Теоретическая физика, т. 2, Теория поля, 7-е изд., Наука, М., 1988, 510 с.
补充文件
