Light cone coordinates and gauge

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

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

 

 

 

encyclopedia

• http://en.wikipedia.org/wiki/Light_cone_gauge
• http://en.wikipedia.org/wiki/
• http://www.scholarpedia.org/

 

 

articles

 

•  
• http://www.ams.org/mathscinet
• [1]http://www.zentralblatt-math.org/zmath/en/
• [2]http://arxiv.org/

• http://dx.doi.org/