# Differentiable and Complex Dynamics of Several Variables

• Pei-Chu Hu
• Chung-Chun Yang
Book

Part of the Mathematics and Its Applications book series (MAIA, volume 483)

1. Front Matter
Pages i-ix
2. Pei-Chu Hu, Chung-Chun Yang
Pages 1-37
3. Pei-Chu Hu, Chung-Chun Yang
Pages 39-62
4. Pei-Chu Hu, Chung-Chun Yang
Pages 63-97
5. Pei-Chu Hu, Chung-Chun Yang
Pages 99-136
6. Pei-Chu Hu, Chung-Chun Yang
Pages 137-177
7. Pei-Chu Hu, Chung-Chun Yang
Pages 179-202
8. Pei-Chu Hu, Chung-Chun Yang
Pages 203-232
9. Pei-Chu Hu, Chung-Chun Yang
Pages 233-274
10. Pei-Chu Hu, Chung-Chun Yang
Pages 275-318
11. Back Matter
Pages 319-341

### Introduction

The development of dynamics theory began with the work of Isaac Newton. In his theory the most basic law of classical mechanics is f = ma, which describes the motion n in IR. of a point of mass m under the action of a force f by giving the acceleration a. If n the position of the point is taken to be a point x E IR. , and if the force f is supposed to be a function of x only, Newton's Law is a description in terms of a second-order ordinary differential equation: J2x m dt = f(x). 2 It makes sense to reduce the equations to first order by defining the velo city as an extra n independent variable by v = :i; = ~~ E IR. . Then x = v, mv = f(x). L. Euler, J. L. Lagrange and others studied mechanics by means of an analytical method called analytical dynamics. Whenever the force f is represented by a gradient vector field f = - \lU of the potential energy U, and denotes the difference of the kinetic energy and the potential energy by 1 L(x,v) = 2'm(v,v) - U(x), the Newton equation of motion is reduced to the Euler-Lagrange equation ~~ are used as the variables, the Euler-Lagrange equation can be If the momenta y written as . 8L y= 8x' Further, W. R.

### Keywords

analysis on manifolds differential equation differential geometry dynamical systems global analysis integration manifold partial differential equation

#### Authors and affiliations

• Pei-Chu Hu
• 1
• Chung-Chun Yang
• 2
1. 1.Shandong UniversityJinanChina
2. 2.The Hong Kong University of Science and TechnologyKowloonHong Kong

### Bibliographic information

• DOI https://doi.org/10.1007/978-94-015-9299-4
• Publisher Name Springer, Dordrecht
• eBook Packages
• Print ISBN 978-90-481-5246-9
• Online ISBN 978-94-015-9299-4
• Buy this book on publisher's site
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