grate by pal EA EA d a+ea-d PSu(l 0,V6u/6a(0) dx du P=EA L Extension to more dimensions OF OF a/ aF OF Build Using divergence theorem aF a/ OF OF duidv+ dinds L Extremum of functional I is obtained when SI=0 or when aF 0/ OF Sui=o on s OF The boxed expressions constitute the Euler-Lagrange equations correspond ing to the variational problem of finding an extremum of the functional I 6� � � � � � � � � � � � � � � � � � � �� � Integrate by parts � d δΠ = dx � L EA − d dx EA δu dx − Pδu(L) du dxδu du dx EA dx + EA L 0 d dx du dx du dx = − δu δu − Pδu(L) 0 Setting δΠ = 0, ∀ δu / δu(0) = 0: d dx � EAdu dx � = 0 du � P = EA � dx L Extension to more dimensions I = F(xi, ui, ui,j )dV V �∂F V dV ∂F δui,j ∂ui,j � ∂F δI = δui + ∂ui = V � ∂F δui + ∂ui � ∂F δui ∂ui,j δui dV ∂ ∂xj ∂ ∂xj − ∂ui,j Using divergence theorem: δI = V � ∂F ∂ui − S � ∂F ∂ui,j δuidV + ∂ ∂xj ∂F δuinjdS ∂ui,j Extremum of functional I is obtained when δI = 0, or when: ∂F � ∂F � ∂ ∂xj − ∂ui,j = 0 , and ∂ui δui = 0 on Su ∂F nj onS − Su = St ∂ui,j The boxed expressions constitute the Euler-Lagrange equations correspond￾ing to the variational problem of finding an extremum of the functional I. 6