Calculus of Variations
Problem I
Obtain the Euler-Lagrange equation that governs the stationary function u(x) of the functional
Z 1
(u′ (x))2 + u(x)u′(x) + (u(x))2 dx
I(u) =
0
Assume that the values of u(x) at the boundaries are given: u(0) = u0 , u(1) = u1 .
Problem II
Obtain the Euler-Lagrange equation that governs the stationary function u(x) of the functional
Z 1
(u′ (x))2 + cos(u(x)) dx
I(u) =
0
Assume that the values of u(x) at the boundaries are given: u(0) = u0 , u(1) = u1 .
Problem III
Obtain the Euler-Lagrange equation that governs the stationary function u(x) of the functional
Z 1
x(u′ (x))2 − u(x)u′ (x) + u(x) dx
I(u) =
0
Assume that the values of u(x) at the boundaries are given: u(0) = u0 , u(1) = u1 .
Problem IV
Determine the stationary function u(x) for the functional
Z 2
I(u) = u′ (x)(1 + x2 u′ (x))dx
1
with the boundary conditions u(1) = 0, u(2) = 1.
Problem V
Determine the stationary function u(x) for the functional
Z 1
(u′ (x))2 + xu′ (x) dx
I(u) =
0
with the boundary conditions u(0) = 0, u(1) = 0.
Problem VI
Determine the stationary function u(x) for the functional
π
1
Z
2
(u(x))2 − (u′ (x))2 dx
I(u) =
2 0
with the boundary conditions u(0) = 0, u( π2 ) = 1.
Problem VII
Determine the stationary function u(x) for the functional
1
1
Z
x2 (u′ (x))2 + 2(u(x))2 dx
I(u) =
2 0
with the boundary conditions u(0) = 0, u(1) = 2.
Problem VIII
Determine the natural boundary conditions at x = 0 and x = 1 for the functional
Z 1
(u′ (x))2 − 2u′ (x) − 2xu(x) dx
I(u) =
0
Problem IX
Determine the stationary function u(x) for the functional
Z 1
(u′ (x))2 − 2u(x)u′(x) − 2u′(x) dx
I(u) =
0
with the essential boundary condition u(1) = 1 and a natural boundary condition at x = 0.
Problem X
Determine the stationary function u(x) for the functional
Z 1
1 ′ 2 2 ′
I(u) = (u (x)) − x u (x) dx
0 2
with the essential boundary condition u(0) = 0 and a natural boundary condition at x = 1.
Problem XI
Determine the stationary function u(x) for the functional
Z π
2
(u′ (x))2 + (u(x))2 − 2u(x) dx
I(u) =
0
with the essential boundary condition u(0) = 0 and a natural boundary condition at x = π2 .
Problem XII
Determine the stationary function u(x) for the functional
1
1
Z
(u′ (x))2 + (u(x))2 − 2xu(x)u′ (x) dx
I(u) =
2 0
with the essential boundary condition u(0) = 1 and a natural boundary condition at x = 1.
Problem XIII
Determine the stationary function u(x) for the functional
1 1
Z
(u′(x))2 + 2xu(x) + 2 dx
I(u) =
2 0
with the essential boundary condition u(0) = 1 and a natural boundary condition at x = 1.
Problem XIV
Show that no continuous stationary function u(x) exists for the functional
Z 1
(u(x))2 + xu(x) − 2(u(x))2 u′ (x) dx
I(u) =
0
with the boundary conditions u(0) = 1, u(1) = 2.
Problem XV
Determine the stationary function u(x) for the dual functional
1 1
Z
(u′(x))2 dx
′
I(u) = u(1) + u(0)u (0) +
2 0
Problem XVI
Determine the stationary function u(x) for the dual functional
1 1
Z
(u′ (x))2 − (u(x))2 − 2u(x) dx
I(u) = −u(1) +
2 0
Problem XVII
Determine the stationary function u(x) for the dual functional
1 π
Z
(u′ (x))2 − (u(x))2 + 2u(x) dx
′
I(u) = −u(π)u (π) +
2 0
Problem XVIII
Determine the stationary function u(x) for the dual functional
1 1 1
Z
2
(u′(x))2 + (u(x))2 dx
I(u) = (u(1)) − u(1) +
2 2 0
Problem XIX
Convert the differential equation
u′′ (x) − u(x) = 0 on the interval [0, 1]
with the boundary conditions u(0) = 0, u(1) = 0 into its equivalent variational form δI = 0
(determine the functional I(u)).
Problem XX
Obtain the Euler-Lagrange equation for the functional
ZZ 2 2 !
∂u ∂u
I(u) = a(x, y) + b(x, y) − c(x, y)(u(x, y))2 dxdy
A ∂x ∂y
with Dirichlet boundary conditions on the whole boundary.
Problem XXI
Consider the functional
ZZZ 2 2 2 !
∂u ∂u ∂u
I(u) = + + + 2f (x, y, z)u(x, y, z) dxdydz
V ∂x ∂y ∂z
with Dirichlet boundary conditions on the whole boundary; here, f (x, y, z) is a given function.
Show that the stationary function u(x, y, z) must satisfy the Poisson equation.
Problem XXII
Obtain the Euler-Lagrange equation for the functional
1 2π 2 2 !
1 1
Z Z
∂u ∂u
I(u) = + 2 rdθdr
2 0 0 ∂r r ∂θ
with the boundary condition u(1, θ) = u0 (θ), where u0 (θ) is specified.
Problem XXIII
The total (kinetic and potential) energy of a rotating elastic bar of unit length is given by the
functional Z t2 Z 1 2 2 !
∂u ∂u
I(u) = + ω 2(r + u(r, t))2 − K 2 drdt
t1 0 ∂t ∂r
where u(r, t) is the radial displacement, ω is the constant angular velocity, and K is a constant.
The bar is fixed at r = 0: u(0, t) = 0 for any t. Obtain the Euler-Lagrange equations (that is, the
equations of motion), along with the boundary conditions at r = 0 and r = 1.
Problem XXIV
Consider the transverse deflection of a string of length ℓ with mass per unit length ρ subject to a
transverse load f (x, t) and constant tension force P . The governing differential equation is
∂2u ∂2u
ρ − P − f (x, t) = 0
∂t2 ∂x2
where u(x, t) is the transverse deflection of the string. Determine the associated energy functional.
Problem XXV
Consider the functional
Z 1
(u′ (x))2 + (v ′ (x))2 − 2u(x)v(x) + 2xu(x) dx
I(u, v) =
0
Determine the Euler-Lagrange equations for u(x) and v(x). Assume that the values of u(x) and
v(x) at the boundaries are specified (Dirichlet boundary conditions).
Problem XXVI
Consider the functional
1
1
Z
(u′ (x))2 + (v ′ (x))2 + 2u′(x)v ′ (x) dx
I(u, v) =
2 0
with the essential boundary conditions u(0) = 0, v(1) = 1, and natural boundary conditions for all
unprescribed conditions. Obtain the Euler-Lagrange equations and all natural boundary conditions
for u(x) and v(x).
Problem XXVII
Determine the stationary functions u(x) and v(x) for the functional
Z 1
(u′ (x))2 + (v ′(x))2 + u′ (x)v ′ (x) dx
I(u, v) =
0
with the essential boundary conditions u(0) = 0, u(1) = 1, v(0) = 2, v(1) = 0.
Problem XXVIII
Determine the natural boundary condition at x = 1 for the functional
Z 1
2
I(u) = (u′ (x)) dx
0
subject to the boundary condition u(0) = 0 and the constraint
Z 1
(u(x))2 dx = 1
0
Problem XXIX
Determine the natural boundary conditions for the functional
Z 1
(u′(x))2 − 4(u(x))2 dx
I(u) =
0
subject to the constraint Z 1
(u(x))2 dx = 1
0
Problem XXX
Determine the natural boundary conditions for the functional
Z 2
x2 (u′ (x))2 − 4(u(x))2 dx
I(u) =
1
subject to the constraint Z 2
(u(x))2 dx = 1
1
Problem XXXI
Determine the Euler-Lagrange equation satisfied by the stationary function u(x) for the functional
Z 1
√
( xu′ (x))2 − (xu(x))2 dx
I(u) =
0
subject to the boundary conditions u(0) = 0, u(1) = 0, and the constraint
Z 1
(xu(x))2 dx = 1
0
Problem XXXII
Determine the Euler-Lagrange equation satisfied by the stationary function u(x) for the functional
Z 1
x(u(x))2 − x2 (u′ (x))2 dx
I(u) =
0
subject to the boundary conditions u(0) = 0, u(1) = 0, and the constraint
Z 1
x(u(x))2 dx = 1
0
Problem XXXIII
Determine the stationary function u(x) for the functional
Z 1
2
I(u) = (u′ (x)) dx
0
subject to the boundary conditions u(0) = 0, u(1) = 0, and the constraint
Z 1
u(x)dx = 1
0
Problem XXXIV
Determine the stationary function u(x) for the functional
Z π
2
I(u) = (u′(x)) dx
0
subject to the boundary conditions u(0) = 0, u(π) = 0, and the constraint
Z π
(u(x))2 dx = 1
0
Problem XXXV
Determine the stationary function u(x) for the functional
1 ∞
Z
(u′ (x))2 − 2xu(x)u′ (x) dx
I(u) =
2 0
subject to the boundary conditions u(0) = 1, u(x) → 0 as x → ∞, and the constraint
1 ∞ 1
Z
(u(x))2 dx =
2 0 4
Problem XXXVI
Determine the Euler-Lagrange equations for the functional
Z 1
(u′ (x))2 + (v ′ (x))2 dx
I(u, v) =
0
subject to the constraint u(x) + (v(x))2 = 2.
Problem XXXVII
Determine the Euler-Lagrange equations for the functional
Z 1
(u′ (x))2 + u(x) + v ′ (x) dx
I(u, v) =
0
subject to the constraint u′ (x) + (v ′ (x))2 = 1.
Problem XXXVIII
Determine the stationary functions u(x) and v(x) for the functional
Z 1
(u′ (x))2 + (v ′ (x))2 − 4xv ′ (x) − 4v(x) dx
I(u, v) =
0
subject to the boundary conditions u(0) = 0, u(1) = 1, v(0) = 0, v(1) = 1, and the constraint
Z 1
(u′ (x))2 − xu′ (x) − (v ′ (x))2 dx = 2
0
Problem XXXIX
Determine the stationary functions u(x) and v(x) for the functional
Z 1
(v(x))2 − 2(u(x))2 dx
I(u, v) =
0
subject to the essential boundary condition u(0) = 1, a natural boundary condition at x = 1, and
the constraint u′ (x) − u(x) − v(x) = 0.
Problem XL
Consider the functional
1
1
Z
(u′(x))2 − (v ′ (x))2 dx
I(u, v) =
0 2
subject to the constraint v (x) + u(x) = 0. Assuming that the end conditions for u(x) and v(x)
′′
are specified at both x = 0 and x = 1, determine the general form of the stationary functions u(x)
and v(x).
Problem XLI
Determine the stationary functions u(x) and v(x) for the dual functional
Z π
1 2 1 ′ 2 ′ 2
I(u, v) = (u(π)) + (u (x)) + (v (x)) + 1 dx
2 0 2
subject to the boundary conditions u(0) = 0, v(0) = 0, u(π) + v(π) = 1, and the constraint
u′ (x) − v(x) = 0.
Hint: Because this is a dual functional, impose the third boundary condition as a constraint.
