Final Exam M 5500, Spring 2018

Solve any six problems

Return at Monday, April 30, before 5 pm

1. Find Euler equation and natural boundary condition, check Legendre condition for the isoperimetric problem

I(u) = min_{u(x)} ∫_{0}^{1} ((u′)^{2} − 3u u′ + 3 sin(x)u^{2}) dx if ∫_{0}^{1} (u) dx = 1

2. Two equal gravitating masses m = 1 with coordinates r_{1}(t) and r_{2}(t)

move in a plane,

r_{i}(t) = (x_{i}(t), y_{i}(t))

Gravitational force is:

F = γ \frac{r_{1} − r_{2}}{|r_{1} − r_{2}|^{3}}

Derive kinetic and potential energy, formulate a variational problem. Find Lagrangian, Hamiltonian, and equations of motion of the masses. Find invariants to the movement.

Hint: Express coordinates through the center of masses r_{0} = \frac{1}{2}(r_{1}+r_{2}) and deviation from the center.

3. Consider the problem

I = min_{u(x)} ∫_{0}^{1}[a (u′)^{2} − c u(u − 1)]dx, u(0) = 0

a. Write Hamiltonian and dual Lagrangian. b. Derive Euler equations and boundary conditions for the primary and the dual problems, derive system of first-order equation and boundary conditions.

4. A process is described by a differential equation

−au′′ + cu^{3} + 1 = 0, in (0, 1) u(0) = 0, u(1) + u′(1) = 3

Write a variational problem (Lagrangian and the boundary term) for which this equation is the Euler equation.

5. Find an approximate solution to the problem

I(u) = min_{u(x)} ∫_{0}^{1} (x(u′ − 1)^{2} + 3x^{2}u^{2}) dx u(0) = u(1) = 0

assuming a representation u(x) = x(1 − x)(A_{0} + A_{1}x + A_{2}x^{2}). Find the constants A_{0}, A_{1}, A_{2}. You may use Maple or similar software.

6. Show that the Euler equation of the problem

inf_{u(x)} ∫_{0}^{1} ((u − x)^{2} + min {(u′ − 1)^{2}, (u′ + 1)^{2}}}) dx, u(0) = 0

does not satisfy the Weierstass test. Describe a minimizing sequence. Find the relaxed formulation of the problem.

7. Find Euler-Lagrange equation and natural boundary conditions for the problem

I(n) = min_{u(x)} [∫_{Ω} ((∇u)^{2} − q u) dx + ∮_{Γ} ((c u + d u^{2}) ds] , ∫_{Ω} u^{2} dx = 1

where x = (x_{1}, x_{2}), Ω is a bounded domain in R_{2} with the smooth boundary Γ, u(x) is a scalar minimizer, q, c, d are parameters.

Extra-Credit Problem

Show that the Euler equation of the problem

inf_{u(x)} ∫_{0}^{1} ((u − αx)^{2} + min {(u′ − 1)^{2}, (u′ + 1)^{2}}}) dx, u(0) = 0

does not satisfy the Weierstass test for certain values of α. Find these values. Describe a minimizing sequence. Find the relaxed formulation of the problem.