Mathematical Tools for Physics - Department of Physics - University

16—Calculus of Variations 405

Exercises

1 For the functionalF[x] =x(0) +

∫π

0 dt

(

x(t)^2 +x ̇(t)^2

)

and the functionx(t) = 1 +t^2 , evaluate

F[x].

2 For the functionalF[x] =

∫ 1

0 dtx(t)

(^2) with the boundary conditionsx(0) = 0andx(1) = 1, what

is the minimum value ofFand what functionxgives it? Start by drawing graphs of variousxthat

satisfy these boundary conditions. Is there any reason to require thatxbe a continuous function oft?

3 With the functionalFof the preceding exercise, what is the functional derivativeδF/δx?