Editing Dirac delta function (section)

===Structural mechanics===
The delta function can be used in [[structural mechanics]] to describe transient loads or point loads acting on structures. The governing equation of a simple [[Harmonic oscillator|mass–spring system]] excited by a sudden force [[impulse (physics)|impulse]] {{mvar|I}} at time {{math|1=''t'' = 0}} can be written

<math display="block">m \frac{d^2 \xi}{dt^2} + k \xi = I \delta(t),</math>

where {{mvar|m}} is the mass, {{mvar|ξ}} is the deflection, and {{mvar|k}} is the [[spring constant]].

As another example, the equation governing the static deflection of a slender [[beam (structure)|beam]] is, according to [[Euler–Bernoulli beam equation|Euler–Bernoulli theory]],

<math display="block">EI \frac{d^4 w}{dx^4} = q(x),</math>

where {{mvar|EI}} is the [[bending stiffness]] of the beam, {{mvar|w}} is the [[deflection (engineering)|deflection]], {{mvar|x}} is the spatial coordinate, and {{math|''q''(''x'')}} is the load distribution. If a beam is loaded by a point force {{mvar|F}} at {{math|1=''x'' = ''x''<sub>0</sub>}}, the load distribution is written

<math display="block">q(x) = F \delta(x-x_0).</math>

As the integration of the delta function results in the [[Heaviside step function]], it follows that the static deflection of a slender beam subject to multiple point loads is described by a set of piecewise [[polynomial]]s.

Also, a point [[bending moment|moment]] acting on a beam can be described by delta functions. Consider two opposing point forces {{mvar|F}} at a distance {{mvar|d}} apart. They then produce a moment {{math|1=''M'' = ''Fd''}} acting on the beam. Now, let the distance {{mvar|d}} approach the [[Limit of a function|limit]] zero, while {{mvar|M}} is kept constant. The load distribution, assuming a clockwise moment acting at {{math|1=''x'' = 0}}, is written

<math display="block">\begin{align}
q(x) &= \lim_{d \to 0} \Big( F \delta(x) - F \delta(x-d) \Big) \\[4pt]
&= \lim_{d \to 0} \left( \frac{M}{d} \delta(x) - \frac{M}{d} \delta(x-d) \right) \\[4pt]
&= M \lim_{d \to 0} \frac{\delta(x) - \delta(x - d)}{d}\\[4pt]
&= M \delta'(x).
\end{align}</math>

Point moments can thus be represented by the [[derivative]] of the delta function. Integration of the beam equation again results in piecewise [[polynomial]] deflection.