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

\(T\) 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
integral of Lagrangian
\(\mathcal{S} = \int L\, \mathrm{d}t\)
this describes the 'total amount that happend' from one moment to another as a particle traces out a path
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
\(\mathcal{S} = \int_{t_0}^{t_1} L(q,\dot{q}) \,dt\)
in quantum mechanics, \(e^{iS/\hbar}\) will describe the 'change in phase' of a quantum system as it traces out a path

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

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/