Wheeler, Mechanics

We don’t usually need a specific sequence to solve problems using delta functions, because they are only
meaningful under integral signs and they are very easy to integrate using their defining property. In physics
they are extremely useful for making “continuous” distributions out of discrete ones. For example, suppose
we have a point charge Q located at x 0 = (x 0 , y 0 , z 0 ) . Then we can write a corresponding charge density as
ρ (x) = Qδ (x − x 0 ) = Qδ (x − x 0 ) δ (y − y 0 ) δ (z − z 0 )
Perform the following integrals over Dirac delta functions:
1. ∫ f(x)δ (x − x 0 ) dx
2. ∫ f(x)δ (ax) dx (Hint: By integrating over both expressions, show that δ(ax) = 1 aδ(x). Note that to
integrate δ(ax) you need to do a change of variables.)
3. Evaluate ∫ f(x)δ (n) (x) dx where
δ (n) (x) = dn
dx n δ (x)
4. ∫ f(x)δ ( x 2 − x 2 0)
dx (This is tricky!)
Show that
Show that
√ m
δ(x) = lim
m→∞ 2π e− m2 x 2
2
m
δ(x) = lim
m→∞ 2 e−m|x|
Let f (x) be a differentiable function with zeros at x 1 , . . . , x n . Assume that at any of these zeros, the
derivative f ′ (x i ) is nonzero. Show that
δ (f (x)) =
n∑
i=1
1
|f ′ (x i )| δ (x − x i)
The mass density of an infinitesimally thin, spherical shell of radius R may be written as
ρ(r, θ, ϕ) =
M δ(r − R)
4πR2 By integrating ρ over all space, find the total mass of the shell.
Write an expression for the mass density of an infinitesimally thin ring of matter of radius R, lying in
the xy plane.
The definition at last! Using the idea of a sequence of functions and the Dirac delta function, we can
now extract the variation. So far, we have defined the variation as
( )∣ d
∣∣∣α=0
δf [x (λ)] ≡
dα f [x(λ, α)] (11)
where x (λ, α) = x (λ) + αh (λ) , and shown that the derivative on the right reduces to
δf [x (λ)] =
∫ 1
( ∂L (x (λ))
− d ∂L (x (λ))
0 ∂x dλ ∂x (1)
+ . . . + (−1) n d n
)
∂L (x (λ))
dλ n h (λ) dλ (12)
∂x (n)
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