"Light cone coordinates and gauge"의 두 판 사이의 차이

2012년 8월 26일 (일) 12:20 판

introduction

light cone gague

1차원에서의 일반해

\(\frac{\partial^2 Y}{\partial t^2}=v^2\frac{\partial^2 Y}{\partial x^2}\) 또는 \(\mu\frac{\partial^2 Y}{\partial t^2}=T\frac{\partial^2 Y}{\partial x^2}\) (\(v=\sqrt{\frac{T}{\mu}}\))
일반해는 \(Y=f(x+vt)+g(x-vt)\)로 주어진다
f는 왼쪽, g는 오른쪽으로 이동하는 파동이며, Y는 그 중첩으로 주어진다

(증명)

\(u=x+at\), \(v=x-at\)라 두자.

그러면 \(Y=f(u)+g(v)\)로 쓸 수 있다.

\(\frac{\partial Y}{\partial t}=\frac{\partial Y}{\partial u}\frac{\partial u}{\partial t} +\frac{\partial Y}{\partial v}\frac{\partial v}{\partial t}=f'(u)a+g'(v)(-a)=af'(u)-ag'(v)\)

\(W(u,v)=\frac{\partial Y}{\partial t}=af'(u)-ag'(v)\).

\(\frac{\partial^2 Y}{\partial t^2}=\frac{\partial W}{\partial t}=\frac{\partial W}{\partial u}\frac{\partial u}{\partial t} +\frac{\partial W}{\partial v}\frac{\partial v}{\partial t}=af''(u)a-ag''(v)(-a)=a^2(f''(u)+g''(v))\)

\(\frac{\partial Y}{\partial x}=\frac{\partial Y}{\partial u}\frac{\partial u}{\partial x} +\frac{\partial Y}{\partial v}\frac{\partial v}{\partial x}=f'(u)+g'(v)\)

\(Z(u,v)=\frac{\partial Y}{\partial x}=f'(u)+g'(v)\)

\(\frac{\partial^2 Y}{\partial x^2}=\frac{\partial Z}{\partial x}=\frac{\partial Z}{\partial u}\frac{\partial u}{\partial x} +\frac{\partial Z}{\partial v}\frac{\partial v}{\partial x}=f''(u)+g''(v)\)

따라서

\(\frac{\partial^2 Y}{\partial t^2}=a^2\frac{\partial^2 Y}{\partial x^2}=a^2(f''(u)+g''(v))\)■

history

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related items

encyclopedia

http://en.wikipedia.org/wiki/Light_cone_gauge
http://en.wikipedia.org/wiki/
http://www.scholarpedia.org/
Princeton companion to mathematics(Companion_to_Mathematics.pdf)

books

[[4909919|]]

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question and answers(Math Overflow)

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http://arxiv.org/