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

 

 

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
  

 

 

 

action

• functional which takes a trajectory(history or path) to a number
  \(\mathcal{S} = \int L\, \mathrm{d}t\)
  
• applying Hamilton's action principle gives rise to a equation of motion
  \({\partial L\over\partial q} - {\mathrm{d}\over \mathrm{d}t }{\partial L\over\partial \dot{q}} = 0\)
  
• mass particle
  \(L(q,\dot{q})=T-V=\frac{1}{2}m{\dot{q}}^2-V(q)\)
  \({\partial L\over\partial q} - {\mathrm{d}\over \mathrm{d}t }{\partial L\over\partial \dot{q}} = 0\) becomes 
  \(\mathcal{S} = \int_{t_0}^{t_1} L(q,\dot{q}) \,dt\)
  

 

 

Euler-Lagrange equation

• if field satisfies the equation of motion, EL is satisfied
  \(\partial_\mu \left( \frac{\partial \mathcal{L}}{\partial ( \partial_\mu \psi )} \right) - \frac{\partial \mathcal{L}}{\partial \psi} = 0.\)
  

 

 

equation of continuity

• current density \(J_{\mu}=(J_0,J_1,J_2,J_3)\) satisfies
  \(\partial^{\mu} J_{\mu}=0\)
  
• we get a conserved quantity
  \(G=\int_V J_0(x) \,d^3 x\)
  
• Lagrangian can be used to express the current density explicity

 

 

currents

• quantum analogues of the conserved densities arising by Noether's theorem
• due to the close relation to observable quantities, they behave similarly to free fields forming the current algebra

 

 

 

 

Lagrangian mechanics

From Lagrangian we obtain the conjugate momentum variable

 

 

Hamiltonian mechanics

conjugate variables are on the equal footing

 

 

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

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

 

 

 

 

canonically conjugate momentum

 

 

 

 

 

http://www.astro.caltech.edu/~golwala/ph106ab/ph106ab_notes.pdf

 

 

history

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

 

 

related items

• Electromagnetism
• sympletic geometry
• integrable Hamiltonian systems and solvable models

 

 

encyclopedia

• http://ko.wikipedia.org/wiki/
• 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.djvu
  
  • V.I. Arnold

• 2010년 books and articles
  
• http://gigapedia.info/1/
• http://gigapedia.info/1/
• http://www.amazon.com/s/ref=nb_ss_gw?url=search-alias%3Dstripbooks&field-keywords=

 

 

articles

 

• http://www.ams.org/mathscinet
• http://www.zentralblatt-math.org/zmath/en/
• http://pythagoras0.springnote.com/
• http://math.berkeley.edu/~reb/papers/index.html
• http://front.math.ucdavis.edu/search?a=&t=&c=&n=40&s=Listings&q=
• http://www.ams.org/mathscinet/search/publications.html?pg4=AUCN&s4=&co4=AND&pg5=TI&s5=&co5=AND&pg6=PC&s6=&co6=AND&pg7=ALLF&co7=AND&Submit=Search&dr=all&yrop=eq&arg3=&yearRangeFirst=&yearRangeSecond=&pg8=ET&s8=All&s7=
• http://dx.doi.org/

 

 

question and answers(Math Overflow)

• http://mathoverflow.net/search?q=
• http://mathoverflow.net/search?q=

 

 

blogs

• 구글 블로그 검색
  
  • http://blogsearch.google.com/blogsearch?q=
  • http://blogsearch.google.com/blogsearch?q=

 

 

experts on the field

• http://arxiv.org/

 

 

links

• Detexify2 - LaTeX symbol classifier
• 수식표 현 안내
• The On-Line Encyclopedia of Integer Sequences
• http://functions.wolfram.com/