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3 Sufficient Conditions of Extrema of Functionals
usually is called the classical theory in the calculus of variations or classical
variational methods. The classical variational method is the study on the extremum
of a functional in the domain.
When y = y(x) is an extremal curve, y
= p, at the moment E = 0.
Let y = y(x) be the extremal curve of the functional (3.3.1) satisfying the
boundary condition (3.3.2). If for all points (x, y) near the extremal curve and the
value y
of the slope function p(x, y) near the extremal curve field, there is
E(x, y, y
, p) ≥ 0 (or ≤ 0)
(3.3.11)
then the inequality (3.3.11) is called the Weierstrass weak condition.
If for all points (x, y) near the extremal curve and any value y
, the inequality
(3.3.11) is true, then the inequality is called the Weierstrass strong condition.
The Weierstrass weak condition and Weierstrass strong condition are collectively
called the Weierstrass condition.
Example 3.3.1 Find the Weierstrass function of the functional
J [y] =
x 1
0
y
3 dx, y(0) = 0, y(x 1 ) = y 1 , x 1 > 0, y 1 > 0
and discuss whether the Weierstrass condition is true.
Solution In the problem the Euler equation of the functional is y
= 0. The extremal
curve satisfying the boundary condition is
y =
y 1
x 1
x
Moreover, the Jacobi equation of the given functional is u
= 0, the general
solution is
u = c 1 x + c 2
From the boundary conditions u(0) = 0, u
(0) = 1 of the Jacobi equation, we
get c 1 = 1, c 2 = 0, so that
u = x
Obviously when 0 < x < x 1 , u = x = 0, so the extremal curve y =
y 1
x 1
x satisfies
the Jacobi condition.
The Weierstrass function is