I have seen wikipedia use the product rule and maybe the chain rule for the variation of the Langragin as follows:
δ[f(g(x,˙x))h(x,˙x)]δx=(δ[f(g)]δgδ[g(x,˙x)]δx)h(x,˙x)+f(g(x,˙x))δ[h(x,˙x)]δx
Does the variation of the Lagrangian satisfy the product rule and chain rule of the derivative?
Answer
OP considers the 'same-time' functional derivative (FD) δf(t)δx(t) := ∂f(t)∂x(t)−ddt∂f(t)∂˙x(t)+….
Here f(t) is shorthand for the function f(x(t),˙x(t),…;t). Although the 'same-time' FD (1) can be notationally useful, it has various fallacies, cf. my Phys.SE answer here.The Leibniz rule δ(f(t)g(t))δx(t) = δf(t)δx(t)g(t)+f(t)δg(t)δx(t)(←Wrong!)
for the 'same-time' FD (1) does not hold. Counterexample: Take f(t)=g(t)=˙x(t).The chain rule δf(t)δx(t) = δf(t)δy(t)δy(t)δx(t)(←Wrong!)
for the 'same-time' FD (1) does not hold. Counterexample: Take f(t)=y(t)2 and y(t)=˙x(t).However, the usual FD δFδx(t) (where F[x] is a functional) does satisfy a Leibniz rule δ(FG)δx(t) = δFδx(t)G+FδGδx(t),
and a chain rule δFδx(t) = ∫dt′ δFδy(t′)δy(t′)δx(t).
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