WOu that satisfy the appropriate differentiability conditions and the homoge- neous essential boundary conditions. Then oF d/OF au dr laul These are the Euler-Lagrange equations corresponding to the variational problem of finding an extremum of the functional I Natural and essential boundary conditions A weaker condition on Ou also allows to obtain the euler equations, we just need OF which is satisfied if du()=0 and Su(b)=0 as before ar (b) ·m(a)=0 and Sut(b)=0 aF (a)=0 and a (b) Essential boundary conditions: ouls=0, or u=uo on Su Natural boundary conditions: on =0 on S Example: Derive Euler's equation corresponding to the total po- tential energy functional II=U+V of an elastic bar of length L, Youngs modulus E, area of cross section A fixed at one end and subject to a load P at the other end EA 2dx Compute the first variation Ea du/du I 6(元)dx-P6u(L) 5� � � ∀δu that satisfy the appropriate differentiability conditions and the homoge￾neous essential boundary conditions. Then: ∂F ∂u − d dx �∂F ∂u� � = 0 These are the Euler-Lagrange equations corresponding to the variational problem of finding an extremum of the functional I. Natural and essential boundary conditions A weaker condition on δu also allows to obtain the Euler equations, we just need: ∂F � b � = 0 ∂u� δu a which is satisfied if: • δu(a) = 0 and δu(b) = 0 as before ∂F • δu(b) = 0 and ∂u�(b) = 0 ∂F • ∂u�(a) = 0 and δu(b) = 0 ∂F ∂F • ∂u�(a) = 0 and ∂u�(b) = 0 Essential boundary conditions: δu� Su = 0, or u = u0 on Su Nat ∂F ural boundary conditions: ∂u� = 0 on S. Example: Derive Euler’s equation corresponding to the total po￾tential energy functional Π = U + V of an elastic bar of length L, Young’s modulus E, area of cross section A fixed at one end and subject to a load P at the other end. � L EA�du�2 Π(u) = dx − Pu(L) 0 2 dx Compute the first variation: EA du �du� δΠ = �2 �2 dxδ dx dx − Pδu(L) 5