Classical field theory and classical mechanics
http://bomber0.myid.net/ (토론)님의 2010년 8월 31일 (화) 14:02 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
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
\(\{f,g\} = \frac{1}{i}[u,v]\)
phase space
canonically conjugate momentum
http://www.astro.caltech.edu/~golwala/ph106ab/ph106ab_notes.pdf
history
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)
blogs
experts on the field