the calculus of variations

Question 1

well-known property of Riemann integrals

Answer 1

if f continuous on interval and non-negative, then integral is non-negative
if f,g continuous on interval and f>g then same inequality for integrals
|integral of f| <= integral of |f|

Question 2

fundamental lemma of the calculus of variations

Answer 2

if v continuous and integral (a to b) v φ dx=0 for all continuous φ with compact support in (a,b) and c2. then v(x)=0

Question 3

satisfies weakly E-L

Answer 3

-integral of partial of L wrt u’ φ’ dx = integral of partial of L wrt u φ dx for all φ in c2 compact

Question 4

extremal in A0

Answer 4

admissible solution (boundary + EL)

Question 5

null Lagrangian

Answer 5

E-L by any function