Advanced Hamiltonian Dynamics and Arnold Diffusion
暫譯: 高階哈密頓動力學與阿諾德擴散
Cheng, Chong-Qing, Xue, Jinxin
- 出版商: Springer
- 出版日期: 2026-07-18
- 售價: $7,160
- 貴賓價: 9.5 折 $6,802
- 語言: 英文
- 頁數: 505
- 裝訂: Hardcover - also called cloth, retail trade, or trade
- ISBN: 9819559642
- ISBN-13: 9789819559640
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相關分類:
工程數學 Engineering-mathematics
海外代購書籍(需單獨結帳)
相關主題
商品描述
This book presents the advanced theory of Hamiltonian dynamics, with an emphasis on the recent development of variational methods and its application to the problem of Arnold diffusion. The main theme of the book is to study the dynamics of finite dimensional nearly integrable Hamiltonian systems, which are small perturbations of integrable systems.
The book consists of two parts. Part I includes the main techniques in Hamiltonian dynamics such as the integrability and nonintegrability theory, the normal form theory (KAM theory, Nekhoroshev theorem), the hyperbolicity theory (the theorem of normally hyperbolic invariant manifold), the variational theory, systems of two degrees of freedom and the connecting orbit theory. In the more advanced Part II, authors specialize to the proof of Arnold diffusion. The techniques in Part I are fully exploited in Part II for people to understand theorems better via applications.The proof the classical Tonelli theorem, some preliminaries of ergodic theory and some basics of genercitity and transversality are given in the Appendix.
This book can be used as the textbook for graduate students in the field of Hamiltonian dynamics. It can also be used as reference for working mathematicians. Before reading this book, obtaining some understanding on Arnold's book Mathematical Methods in Classical Mechanics, algebraic topology, differential topology and convex analysis will be helpful.
商品描述(中文翻譯)
這本書介紹了哈密頓動力學的進階理論，重點在於變分方法的最新發展及其在阿諾德擴散問題上的應用。書中的主要主題是研究有限維度的近可積哈密頓系統的動力學，這些系統是可積系統的小擾動。
本書分為兩部分。第一部分包括哈密頓動力學中的主要技術，如可積性與不可積性理論、標準型理論（KAM 理論、內科羅舍夫定理）、超曲率理論（正常超曲率不變流形的定理）、變分理論、兩自由度系統及連接軌道理論。在更進階的第二部分，作者專注於阿諾德擴散的證明。第一部分中的技術在第二部分中得到了充分的應用，以幫助讀者更好地理解定理。附錄中提供了經典的托內利定理的證明、一些遍歷理論的初步知識以及一般性和橫斷性的基本概念。
這本書可以作為哈密頓動力學領域研究生的教科書，也可以作為在職數學家的參考資料。在閱讀本書之前，對阿諾德的書籍《數學方法在經典力學中的應用》、代數拓撲、微分拓撲和凸分析有一定的理解將會有所幫助。
作者簡介
Chongqing Cheng is a professor at Nanjing University. He has been engaged in the research of dynamical systems for many years. His main research interest is Hamiltonian dynamical systems, including dynamical instability, variational construction of connecting orbits, Arnold diffusion, KAM theory and weak KAM theory. He was an invited speaker at ICM 2010 Hyderabad.
Jinxin Xue is a professor at Tsinghua University. His field of research is dynamical systems and his primary interests are to find various special orbits in dynamical systems. Prof. Xue solved the Painleve conjecture and the Arnold diffusion conjecture (jointly with C-Q. Cheng) in the smooth category for convex systems. Besides Hamiltonian dynamics, he also works on fields including hyperbolic dynamics, group actions, symplectic topology, mean curvature flow, etc.
作者簡介(中文翻譯)
重慶程是南京大學的教授。他從事動力系統的研究已有多年，主要研究興趣為哈密頓動力系統，包括動力不穩定性、連接軌道的變分構造、阿諾德擴散、KAM 理論及弱 KAM 理論。他曾在 2010 年海德拉巴的國際數學家大會（ICM）上擔任特邀演講者。
薛金鑫是清華大學的教授。他的研究領域為動力系統，主要興趣在於尋找動力系統中的各種特殊軌道。薛教授在平滑類別中解決了 Painleve 猜想和阿諾德擴散猜想（與程春青共同合作），針對凸系統進行研究。除了哈密頓動力學外，他還從事包括雙曲動力學、群作用、辛拓撲、平均曲率流等領域的研究。
