Let n ≥ 1 be an integer, let to < t₁ and x0, x₁ be real constant n-vectors, and consider the functional of n dependent variables x1 In: rt1 C[x] = [*^ dt L(t,x,x), x(to) = xo, x(t₁) = x1, where x = to ..., 1, . . ., £n) and x = (±1,...,xn), and where ik (x1,..., k = 1, . . ., n. Show that on a stationary path = dxk for dt n d ᎥᏞ aL L ik == dt Əxk Ət k=1 Hence find a first-integral for the case when L is independent of t.

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