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374 CHAPTER24. THEFEYNMANPATHINTEGRAL

Thisexamplegeneralizesinastraightforwardway

δ

δf(x)

fn(y) = lim!→ 0

(f(y)+!δ(y−x))n−fn(y)

!

= nfn−^1 (y)δ(x−y)

δ

δf(x)

R(f(y)) = lim

!→ 0

R(f(y)+!δ(y−x))−R(f(y))

!

=

∂R(f(y))

∂f(y)

δ(x−y) (24.39)

Functional differentiation isthe easiest and quickest way to derive the Euler-
Lagrangeequationsofmotion,fromthecondition

δS

δx(t)

= 0 (24.40)

thattheactionshouldbestationarywithrespecttoaninfinitesmalchangeofpath.
Thisworksasfollows:

0 =

δS

δx(t)

=

δ

δx(t)

∫

dt′

{

1

2

mx ̇^2 −V(x(t′))

}

=

∫

dt′

{

1

2

m

δ

δx(t)

x ̇^2 −

δ

δx(t)

V(x(t′))

}

(24.41)

Now, usingthe definition of the functionalderivative and the properties of delta
functions,wehave

m

δ

δx(t)

x ̇^2 = lim

!→ 0

[∂t′(x(t′)+!δ(t−t′))]^2 −(∂t′x(t′))^2

!

= 2 x ̇(t′)∂t′δ(t−t′)

= − 2 ∂^2 t′x(t′)δ(t−t′) (24.42)

Then

0 =

∫

dt′

{

−m∂t^2 ′x(t′)δ(t−t′)−

∂V(x(t′))

∂x(t′)

δ(t−t′)

}

= −m∂t^2 x−

∂V

∂x

(24.43)

whichis,ofcourse,Newton’sLawofmotionF=ma.
Togetalittlemorepracticewithfunctionalderivatives,letsfindtheequationsof
motionforawaveonastring. Denotethewavefunctionofthestring(whichinthis