#extremum

Similarly, to find a minimum (maximum) of a functional Φ[f(x)], we take its directional derivative δΦ[f(x),Δ(x)], a derivative in a direction of a function increment Δ(x) that is called VARIATION, and look for such a solution f(x) of a differential equation δΦ[f(x),Δ(x)]=0 that is independent of a direction of a function increment Δ(x).

For functionals like Φ[f(x)] = ∫[a,b] F[x,f(x),f '(x)]dx defined for smooth functions f(x) on segment [a,b] with fixed values at the ends (that is, f(a)=A, f(b)=B for all f(x)) this differential equation is called Euler-Lagrange equation.

A new lecture, where this differential equation is derived for the type of functionals mentioned above, has been just released to

UNIZOR.COM > Physics+ 4 All > Variations > Euler-Lagrange.

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