Consider the functional (a) S[11, 12] = [ dr [7y/² + 24yy⁄2 + 21y2² + 4y{ + 12y192+9y²] . two coupled Euler-Lagrange equations giving the stationary paths of S[y1, y2]. (b) Express the functional S in terms of the two new dependent variables 21, 22, where Y1 and 92 are given in terms of the new dependent variables by Y₁ = azı 322, Y2 = 21 + 222, and where a is a constant. Determine the value of a such that the functional S[21, 22] can be written as the sum of two functionals, S₁[21] + S2[22], where S₁[21] depends only upon 21 and S2[22] depends only on upon z2. Calculate S₁[21] and S2[22] for this value of a. Euler - Lagrange equations for y, and you are dx d (14 y'₁ + 24y² ) - (84, +12 y2) = -༠ and (24y, + 42 y2) - (1zy, + 18y2) = =0 ⇒ 14y" + 24y" - Sy, -12yz =0 and 7 7y," + 12y" -44, - byz D= ⇒ 24 y" + 42 y2 - 124, -18y2=0 and ⇒ 8 y₁" + 14 y" - 441 - 6 y 2 =0 = 0 ⇒ 4y, + 7y₂" -241-342-