(a) Consider the functional S[y, y2] = [ da ((5 y{² + 6y{y2 + 2 y2²) + (32 y² + 28 y1 y2 + 5 y²)) where the limits of in

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A Consider The Functional S Y Y2 Da 5 Y 6y Y2 2 Y2 32 Y 28 Y1 Y2 5 Y Where The Limits Of In 1 (166.63 KiB) Viewed 33 times

(a) Consider the functional S[y, y2] = [ da ((5 y{² + 6y{y2 + 2 y2²) + (32 y² + 28 y1 y2 + 5 y²)) where the limits of integration and boundary conditions have been omitted for clarity. (i) Find the two coupled Euler-Lagrange equations for S[y₁, y2]. Two new dependent variables z₁ and 22 are defined via the equations = - (21+22₂), Y2 = a ²₁ +222. (ii) By substituting for y₁ and y2 in S[y₁, 92], find the value of a so that the induced functional S[21, 22] has the form S[21, 22] = S₁ [21] + S2[22], where S₁ is a functional depending only on 2₁ and S₂ is a functional depending only on 22. (iii) For each of S₁[₁] and S₂[22], solve the corresponding Euler-Lagrange equation to give the general solutions for 2₁ and 22. (iv) Hence, find the general solution to the Euler-Lagrange equations of the original functional S[y₁, y2]. (b) Consider the functional dx F(y, y'), l y(2) = -2, y(6) = 3, S[y] = d 2 13 where F(y, y') = 5 yº y¹³ and y > 0. Calculate the first-integral for S and hence find the stationary path.