"Classical field theory and classical mechanics"의 두 판 사이의 차이

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<h5 style="margin: 0px; line-height: 2em;">canonically conjugate momentum</h5>
 
<h5 style="margin: 0px; line-height: 2em;">canonically conjugate momentum</h5>
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<math>{\partial L\over\partial q} - {\mathrm{d}\over \mathrm{d}t }{\partial L\over\partial \dot{q}} = 0</math>
  
 
 
 
 
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==== 하위페이지 ====
  
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* [[classical field theory  and classical mechanics|classical field theory and classical mechanics]]<br>
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** [[Lagrangian formalism]]<br>
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** [[Legendre transformation]]<br>
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** [[Nonlinear Sigma model]]<br>
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** [[symmetry and conserved quantitiy : Noether's theorem]]<br>
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** [[symplectic geometry]]<br>
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*** [[action-angle variables]]<br>
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*** [[canonical transformation]]<br>
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*** [[Hamiltonian flows]]<br>
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*** [[moment map]]<br>
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*** [[quantization of Poisson algebras]]<br>
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*** [[symplectic leaves]]<br>
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*** [[two-body problem]]<br>
  
 
 
 
 

2012년 6월 10일 (일) 17:17 판

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
  • \(T\) kinetic energy
  • \(V\) potential energy
  • We have Lagrangian \(L=T-V\)
  • Define the Hamiltonian
  • \(H =p\dot q-L\)
  • \(p\dot q\) is twice of kinetic energy
  • Thus the Hamiltonian represents \(H=T+V\) the total energy of the system

 

 

Lagrangian formalism

 

 

 

Hamiltonian mechanics

 

 

 

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

 

 

canonically conjugate momentum

\({\partial L\over\partial q} - {\mathrm{d}\over \mathrm{d}t }{\partial L\over\partial \dot{q}} = 0\)

 

 

하위페이지

 

 

 

links and webpages

 

question and answers(Math Overflow)

 

 

 

history

 

 

related items

 

 

encyclopedia

 

 

books
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