Variational Methods for Engineers with Matlab

6Dirac’s Delta

Dirac’s delta δ₀ (sometimes denoted as δ(x)) is a fundamental linear functional. It has been introduced in examples 3.10, 4.2, 4.3, 4.4, 4.5.

Dirac’s delta is often used to denote instantaneous impulsions, concentrated forces, sources or other quantities having their action concentrated at a point or an instant of time. For instance, Dirac’s delta is used to describe

• 1) An instantaneous variation of the velocity
• 2) A punctual source or a sink
• 3) A punctual charge
• 4) A punctual force

Formally, Dirac’s delta δ₀ is a linear functional, i.e. a function that associates a real number to a function, i.e. δ₀: V→ , where V is a convenient space of functions. For a function ∈ V, we have:

Usually, δ₀ is defined on the space formed by the indefinitely differentiable functions having compact support and taking its values on (see below).

Since variations of energy are physically connected to the action of forces, it is usual to refer to Dirac’s delta δ₀ as Dirac’s delta function δ(x) and, analogously, to use the notation ∫δ(x)φ(x)dx to refer to δ₀(φ). These notations are not mathematically correct, but they are often useful in practice

However, is given by (φ) = φ(x₀) and sometimes ...