Introduction

This course is a brief introduction to Hamiltonian systems and symplectic geometric numerical methods(simply called symplectic methods). The topics include:

(1) Basics of Hamiltonian dynamical systems;

(2) Stability and KAM theory;

(3) Kang Feng’s idea on geometric numerical methods of dynamical systems;

(4) Symplectic methods based on Hamilton-Jacobi equations and generating function theory;

(5) Symplectic methods based on Runge-Kutta\Patitioned Runge-Kutta methods;

(6) Symplectic methods based on the splitting of vector fields and composition of phase flows;

(7) Numerical stability;

(8) Backward error analysis and almost preservation of invariants

Lecturer Intro

Zaijiu Shang is a Professor of the Academy of Mathematics and Systems Science, Chinese Academy of Sciences, and a Post Teacher at the University of Chinese Academy of Sciences (2015-). He was the deputy director (2003-2011) and the director (2012-2016) of the Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences. He has been served as a member of editorial boards of Acta Math. Appl. Sinica (2007-), Acta Math. Sinica (2009-), Science China: Mathematics (2013-), and Applied Mathematics (HUST 2013-). He is working in the fields of dynamical systems and geometrical numerical methods. He won the second prize in “the Science and Technology Progress Award of the State Education Commission (1993)”. He was one of the core members of the project “Symplectic Geometric Algorithms of Hamiltonian Systems” which won the first prize of the National Natural Science Awards (Kang Feng etc., 1997), and his representative achievements include stability theory of symplectic alg