課程資訊
課程名稱
 數學物理專題
Special Topics in Mathematical Physics 
開課學期
 105-1 
授課對象
 理學院  物理學研究所  
授課教師
 陳義裕 
課號
 Phys8133 
課程識別碼
 222 D1990 
班次
  
學分
 3 
全/半年
 半年 
必/選修
 選修 
上課時間
 星期一2,3,4(9:10~12:10) 
上課地點
 新物112 
備註
 總人數上限：60人
外系人數限制：5人 
Ceiba 課程網頁
 http://ceiba.ntu.edu.tw/1051Phys8133_MathTop 
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課程概述

This course introduces to the students some of the frequently used perturbation techniques and asymptotic analysis in physics. It is expected that the students taking this course are well-versed in materials treated in our undergraduate courses of Applied Math I-IV. Familiarity with classical mechanics and quantum mechanics is also assumed. 

課程目標
 Outline of Contents:
(The following is only a rough outline. An * indicates a topic which might be omitted when time does not permit.)

1. Perturbation on roots of polynomials- a warm-up:
 The quadratic equation revisited
 Regular perturbation theory, the way not to go!
 Iteration, a typically much faster converging scheme

2. Perturbation of eigenvalue problems:
 Regular (Rayleigh-Schrodinger) perturbation and the solubility condition
 A complex representation
 Iteration, again
 Degeneracy
 Divergence of the perturbation series and level crossing
 Rayleigh-Ritz method

3. Multiple-scale analysis:
 Resonance and secular behavior
 Two-timing
 Method of averaging
 Action-angle variables
 Adiabatic invariant of classical mechanics
 Periodic perturbation and Floquet theory
 Mathieu’s equation and its solutions

4. Boundary layers
 A model equation exhibiting boundary layers
 Examples from fluid mechanics

5. Boundary perturbations:
 Green function representation
 Perturbing the shape of the boundary
 Perturbation of the boundary conditions

6. Diffraction of scalar waves:
 Green function representation
 Kirchhoff approximation, Fraunhoffer and Fresnel diffraction
 Sommerfeld and Rabinowicz representation

7. Asymptotic expansion of integrals:
 Integral representations. Why?
 Integration by parts
 Laplace’s method and Watson’s lemma
 Method of stationary phase
 Method of steepest descent
 Stokes phenomenon
 Asymptotic evaluation of sums

8. Semiclassical approximation:
 Hamilton-Jacobi theory
 The path integral formulation
 Short wave approximation
 The trace formula
 JWKB approximation

9. Fourier analysis*:
 Gibbs phenomenon and Pinsky Phenomenon
 Relevance to wave optics

10. KAM theory*:
 The small-divisor problem
 Continued fractions, and their applications
 Perturbations do not necessarily destroy all the good stuff
 
課程要求
 Grading Policy:
1. Homework: 60%
2. One final exam: 40%
 
預期每週課後學習時數
  
Office Hours
  
指定閱讀
 None, officially. 
參考書目
 References:
1. Carl M. Bender, Steven A. Orszag , Advanced Mathematical Methods for Scientists and Engineers I: Asymptotic Methods and Perturbation Theory, Springer (1999). NTU campus access:
http://link.springer.com/book/10.1007%2F978-1-4757-3069-2

2. R.S. Johnson, Singular Perturbation Theory, Springer (2005).
NTU campus access:
http://link.springer.com/book/10.1007/b100957
 
評量方式
(僅供參考)
    
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