# Calculus of Variations

### From Exampleproblems

solution Find the path that minimizes the arclength of the curve between and .

solution Find the extrema of subject to the constraint .

solution Find the maximum of subject to the constraint .

solution Write the Euler-Lagrange equations for .

solution Constraint problem: Minimize s.t. .

solution Derive the Euler-Lagrange equation from the attempt to minimize the functional

solution Minimize the functional from classical mechanics:

solution Find the extrema of .

solution Find the extrema of .

solution Show that the first variation satisfies the homogeneity condition .

solution , where is a normed linear space, is linear if and . Which of the following are functionals on are linear?

(a)

(b)

(c)

(d) The set of all continuous functions on satisfying

(e) The set of all continuous functions on satisfying

solution Find the extremal for

solution Compute the first variation of

solution Compute the first variation of

solution Compute the first variation of

solution Minimize

solution Find the extremals of

solution Find the Euler equation for

solution Minimize

solution Minimize

solution Obtain a necessary condition for a function to be a local minimum of the functional

solution Find the Euler equation for the functional

where is a closed region in the plane and has continuous second partial derivatives.

solution Find the extremal of the functional subject to the constraint .

solution Determine the function that minimizes the functional .

solution Let be a functional on a subset of a normed linear space .

(a) Define precisely the first variation of at and admissible .

(b) Show that if exists for a certain admissible , then also exists for every real number , and .

solution Compute the first variation for :

solution Compute the first variation for :

solution Compute the first variation