Girish Mahajan (Editor)

First variation

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In applied mathematics and the calculus of variations, the first variation of a functional J(y) is defined as the linear functional δ J ( y ) mapping the function h to

δ J ( y , h ) = lim ε → 0 J ( y + ε h ) − J ( y ) ε = d d ε J ( y + ε h ) | ε = 0 ,

where y and h are functions, and ε is a scalar. This is recognizable as the Gâteaux derivative of the functional.

Example

Compute the first variation of

J ( y ) = ∫ a b y y ′ d x .

From the definition above,

δ J ( y , h ) = d d ε J ( y + ε h ) | ε = 0 = d d ε ∫ a b ( y + ε h ) ( y ′ + ε h ′ )   d x | ε = 0 = d d ε ∫ a b ( y y ′ + y ε h ′ + y ′ ε h + ε 2 h h ′ )   d x | ε = 0 = ∫ a b d d ε ( y y ′ + y ε h ′ + y ′ ε h + ε 2 h h ′ )   d x | ε = 0 = ∫ a b ( y h ′ + y ′ h + 2 ε h h ′ )   d x | ε = 0 = ∫ a b ( y h ′ + y ′ h )   d x

References

First variation Wikipedia
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