Classical field theory and classical mechanics

introduction

• can be formulated using classical fields and Lagrangian density
• change the coordinates and fields accordingly
• require the invariance of action integral over arbitrary region
• this invariance consists of two parts : Euler-Lagrange equation and the equation of continuity
• three important conserved quantity
  
  • energy
  • momentum
  • angular momentum

 

 

notation

• dynamical variables \(q_{k}, \dot{q}_k\) for \(k=1,\cdots, N\)
  
• \(T\) kinetic energy
  
• \(V\) potential energy
  
• We have Lagrangian \(L=T-V\)
  
• Define the Hamiltonian
  
• \(H =\sum_{k=1}^{N} p_{k}\dot{q}_{k}-L\)
  
• \(p\dot q\) is twice of kinetic energy
  
• Thus the Hamiltonian represents \(H=T+V\) the total energy of the system
  

 

 

Lagrangian formalism

• Lagrangian formalism

 

 

canonically conjugate momentum

• canonically conjugate momenta
  \(p_{k}=\frac{\partial L}{\partial \dot{q}_k}\)
  
• instead of \(q_{k}, \dot{q}_k\), one can use \(q_{k}, p_{k}\) as dynamical variables

 

 

 

Hamiltonian mechanics

• conjugate variables are on the equal footing
  
• 고전역학에서의 가적분성 항목 참조

 

 

 

Poisson bracket

For \(f(p_i,q_i,t), g(p_i,q_i,t)\) , we define the Poisson bracket

\(\{f,g\} = \sum_{i=1}^{N} \left[ \frac{\partial f}{\partial q_{i}} \frac{\partial g}{\partial p_{i}} - \frac{\partial f}{\partial p_{i}} \frac{\partial g}{\partial q_{i}} \right]\)

In quantization we have correspondence

\(\{f,g\} = \frac{1}{i}[u,v]\)

 

 

phase space

 

 

하위페이지

• classical field theory and classical mechanics
  
  • Lagrangian formalism
    
  • Legendre transformation
    
  • Nonlinear Sigma model
    
  • symmetry and conserved quantitiy : Noether's theorem
    
  • symplectic geometry
    
    • action-angle variables
      
    • canonical transformation
      
    • Hamiltonian flows
      
    • moment map
      
    • quantization of Poisson algebras
      
    • symplectic leaves
      
    • two-body problem
      

 

 

 

links and webpages

• [1]http://www.astro.caltech.edu/~golwala/ph106ab/ph106ab_notes.pdf
• Classical Mechanics
  
  • John Baez

 

question and answers(Math Overflow)

• http://mathoverflow.net/search?q=
• http://mathoverflow.net/search?q=
• http://goo.gl/rUJBV

 

 

 

history

• http://www.google.com/search?hl=en&tbs=tl:1&q=

 

 

related items

• Electromagnetism
• Einstein field hequation
• sympletic geometry
• integrable Hamiltonian systems and solvable models

 

 

encyclopedia

• http://en.wikipedia.org/wiki/Classical_field_theory
• http://en.wikipedia.org/wiki/Continuity_equation
• http://en.wikipedia.org/wiki/current_density
• http://en.wikipedia.org/wiki/Noether's_theorem

• http://ko.wikipedia.org /wiki/작용
• http://en.wikipedia.org/wiki/Canonical_coordinates
• http://en.wikipedia.org/wiki/Lagrangian_mechanics
• http://en.wikipedia.org/wiki/Lagrangian
• http://en.wikipedia.org/wiki/poisson_bracket
• http://en.wikipedia.org/wiki/Action_(physics)

• http://en.wikipedia.org/wiki/
• http://en.wikipedia.org/wiki/
• Princeton companion to mathematics(Companion_to_Mathematics.pdf)

 

 

books

•  
  
• Classical mechanics Classical_Mechanics.djvuV.I. Arnold
  
• Emmy Noether’s Wonderful Theorem
  
•  
  Electrodynamics and Classical Theory of Fields and Particles
  

 

 

expositions

• Benci V. Fortunato D., Solitary waves in classical field theory, in Nonlinear Analysis and Applications to Physical Sciences