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

2010년 9월 22일 (수) 17:32 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

Lagrangian formalism

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

canonically conjugate momentum

links and webpages

history

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

related items

encyclopedia

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

articles

question and answers(Math Overflow)

blogs

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

experts on the field

http://arxiv.org/

@@ 30번째 줄: / 30번째 줄: @@
 <h5>Lagrangian formalism</h5>
-* Lagrangian f
+* [[Lagrangian formalism]]
-<h5 style="margin: 0px; line-height: 2em;">action</h5>
-*  functional which takes a trajectory(history or path) to a number<br>
-*  integral of Lagrangian<br><math>\mathcal{S} = \int L\, \mathrm{d}t</math><br>
-*  this describes the 'total amount that happend' from one moment to another as a particle traces out a path<br>
-*  applying Hamilton's action principle gives rise to a equation of motion<br><math>{\partial L\over\partial q} - {\mathrm{d}\over \mathrm{d}t }{\partial L\over\partial \dot{q}} = 0</math><br>
-*  mass particle<br><math>L(q,\dot{q})=T-V=\frac{1}{2}m{\dot{q}}^2-V(q)</math><br><math>{\partial L\over\partial q} - {\mathrm{d}\over \mathrm{d}t }{\partial L\over\partial \dot{q}} = 0</math> becomes <br><math>\mathcal{S} = \int_{t_0}^{t_1} L(q,\dot{q}) \,dt</math><br>
-*  in [[quantum mechanics]], <math>e^{iS/\hbar}</math> will describe the 'change in phase' of a quantum system as it traces out a path<br>
-<h5>Euler-Lagrange equation</h5>
-*  if field satisfies the equation of motion, EL is satisfied<br><math>\partial_\mu \left( \frac{\partial \mathcal{L}}{\partial ( \partial_\mu \psi )} \right) - \frac{\partial \mathcal{L}}{\partial \psi} = 0.</math><br>
-<h5>equation of continuity</h5>
-*  current density <math>J_{\mu}=(J_0,J_1,J_2,J_3)</math> satisfies<br><math>\partial^{\mu} J_{\mu}=0</math><br>
-*  we get a conserved quantity<br><math>G=\int_V J_0(x) \,d^3 x</math><br>
-* Lagrangian can be used to express the current density explicity
-<h5>currents</h5>
-* 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
-<h5 style="margin: 0px; line-height: 2em;">Lagrangian mechanics</h5>
-From Lagrangian we obtain the conjugate momentum variable