Page 70 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

Euler-Lagrange equation 57

2.2.1 Exercises
N
Exercise 2.2.1 Generalize Theorem 2.1 to the case where u :[a, b] → R ,
N ≥ 1.

Exercise 2.2.2 Generalize Theorem 2.1 to the case where u :[a, b] → R and
( )
Z
b ³ ´
(P) inf I (u)= f x, u (x) , ..., u (n) (x) dx
u∈X
a
© ª
n
where X = u ∈ C ([a, b]) : u (j) (a)= α j ,u (j) (b)= β , 0 ≤ j ≤ n − 1 .
j
Exercise 2.2.3 (i) Find the appropriate formulation of Theorem 2.1 when u :
[a, b] → R and
( )
b
Z
0
(P) inf I (u)= f (x, u (x) u (x)) dx
u∈X
a
© ª
1
where X = u ∈ C ([a, b]) : u (a)= α , i.e. we leave one of the end points free.
(ii) Similar question, when we leave both end points free; i.e. when we min-
1
imize I over C ([a, b]).
Exercise 2.2.4 (Lagrange multiplier). Generalize Theorem 2.1 in the fol-
lowing case where u :[a, b] → R,
( )
Z b
0
(P) inf I (u)= f (x, u (x) ,u (x)) dx ,
u∈X
a
( )
b
Z
1
X = u ∈ C ([a, b]) : u (a)= α, u (b)= β, g (x, u (x) ,u (x)) dx =0
0
a
2
where g ∈ C ([a, b] × R × R).
3
Exercise 2.2.5 (Second variation of I).Let f ∈ C ([a, b] × R × R) and
( )
Z b
0
(P) inf I (u)= f (x, u (x) ,u (x)) dx
u∈X a
© 1 ª 2
where X = u ∈ C ([a, b]) : u (a)= α, u (b)= β .Let u ∈ X ∩ C ([a, b]) be a
minimizer for (P). Show that the following inequality
Z b
£ 2 02 ¤
f uu (x, u, u ) v +2f uξ (x, u, u ) vv + f ξξ (x, u, u ) v dx ≥ 0
0
0
0
0
a