String Theory Demystified

CHAPTER 2 Equations of Motion 35
Similarly,
σ µ
P X P X X
µ
µ
σ δ
∂
= δ δ
∂ σ
∂
( )−
∂
∂P
σ µ µ
σ
µ
∂σ
This means that the variation of the action can be written as
τ
τ
τ µ µ µ
δS T dτ dσ P δX δX
τ
µ
τ
τ
P
f ⎡ ∂
∂ ⎤
=− ∫ ∫ ⎢ ( )− ⎥
i 0
⎣⎢
∂ ∂
⎦⎥
⎡ ∂
∂
− T∫ d ∫ d ⎢ ( P X )−X
⎣⎢
∂ ∂
P
σ
τ f
⎤
σ µ µ µ
τ σ δ δ
τ
µ
⎥
i 0 σ
σ
⎦⎥
Recall from classical mechanics that a variation is defi ned such that variation at
the endpoints is 0, that is, at the initial and fi nal times δ µ
X = 0. In the case of the
endpoints of the string, we can apply either Neumann or Dirichlet boundary
conditions so we will have to handle each case differently (more on this as we go
along). For now, let’s take δ µ
X = 0 for simplicity. This means that we can throw
away the terms in the above expression which are integrals of total derivatives:
τ f
τ i
This leaves us with
∂ τ µ
∂ σ µ
dτ( P δX dσ P δX
µ )= 0 ( µ )= 0
∂ τ
0 ∂ σ
∫ ∫
τ
τ ⎛ ∂ ⎞
µ µ
τ
0 = δS = T∫ dτ∫ dσ⎜δX ⎟ +
0
⎝ ∂τ
∫ τ
τ
τ
⎠
P
σ
f
f ⎛ ∂ ⎞
µ µ
T d ∫ dσσ⎜δX
⎟
i
i 0
⎝ ∂σ
⎠
P
τ
∫ ∫
f
= T dτ dσδX τi
0
µ
τ
∂Pµ
+
∂τ
∂
σ ⎛ P ⎞ µ
⎜ ⎟
⎝ ∂σ
⎠
This gives us the equation of motion for the string, derived from the Nambu-Goto
action:
τ σ
∂P
∂P
µ µ
+ = 0 (2.26)
∂ τ ∂σ