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

links and webpages

question and answers(Math Overflow)

http://mathoverflow.net/questions/30886/applications-of-classical-field-theory

history

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

related items

encyclopedia

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