3.5 Sufficient Conditions of Extrema of Functionals
223
3.5.2 The Legendre Sufficient Conditions
Theorem 3.5.4 If the curve y = y(x) (x 0 ≤ x ≤ x 1 ) is the extremal curve of
the functional (3.2.1) satisfying the boundary condition (3.2.2), and it satisfies the
Jacobi condition, on the extremal curve y = y(x), the Legendre condition holds,
namely F y y (x, y, y
) does not change its sign; then the functional (3.2.1) satisfying
the fixed boundary condition (3.2.2) obtains weak extremum on the extremal curve
y = y(x). When F y y > 0, it obtains weak minimum; When F y y < 0, it obtains
weak maximum.
Proof According to the expression (3.3.7) and expression (3.4.5), the increment of
the functional J [y(x)] on the extremal curve y = (x) can be expressed as
J =
x 1
x 0
E(x, y, y
, p)dx =
x 1
x 0
(y
− p)
2
2!
F y y (x, y, q)dx
where, q is between p and y
.
Since supposing on the extremal curve y = y(x), F y y (x, y, y
) = 0, thus
according to the continuity of the functional, for the point near the extremal curve
y = y(x) and the value y
near the slope function p, there is F y y (x, y, y
) = 0,
consequently J = 0. and when F y y > 0, J > 0; When F y y < 0, J < 0.
According to the definition of weak minimum (or weak maximum), the theorem is
proved. Quod erat demonstrandum.
Theorem 3.5.5 If the curve y = (x) (x 0 ≤ x ≤ x 1 ) is the extremal curve of the
functional (3.2.1) satisfying the boundary condition (3.2.2), and it satisfies the Jacobi
condition, for all points (x, y) in a zero-th order neighborhood of the extremal curve
y = (x) and any value y
, the Legendre condition holds, namely F y y (x, y, q) does
not change its sign, and the first order Taylor formula of the function F(x, y, y
) holds
at y
= p; Then the functional (3.2.1) satisfying the fixed boundary condition (3.2.2)
obtains strong extremum on the extremal curve y = (x). When F y y (x, y, q) ≥ 0, it
obtains a strong minimum; When F y y (x, y, q) ≤ 0, it obtains a strong maximum.
Proof According to the given conditions, for any y
, there is
F(x, y, y
) = F(x, y, p) + (y
− p)F p (x, y, p) +
(y
− p)
2
2!
F y y (x, y, q)
where, q is between p and y
.
Substituting the above expression into E-function, we get
E(x, y, y
, p) =
(y
− p)
2
2!
F y y (x, y, q)
223
3.5.2 The Legendre Sufficient Conditions
Theorem 3.5.4 If the curve y = y(x) (x 0 ≤ x ≤ x 1 ) is the extremal curve of
the functional (3.2.1) satisfying the boundary condition (3.2.2), and it satisfies the
Jacobi condition, on the extremal curve y = y(x), the Legendre condition holds,
namely F y y (x, y, y
) does not change its sign; then the functional (3.2.1) satisfying
the fixed boundary condition (3.2.2) obtains weak extremum on the extremal curve
y = y(x). When F y y > 0, it obtains weak minimum; When F y y < 0, it obtains
weak maximum.
Proof According to the expression (3.3.7) and expression (3.4.5), the increment of
the functional J [y(x)] on the extremal curve y = (x) can be expressed as
J =
x 1
x 0
E(x, y, y
, p)dx =
x 1
x 0
(y
− p)
2
2!
F y y (x, y, q)dx
where, q is between p and y
.
Since supposing on the extremal curve y = y(x), F y y (x, y, y
) = 0, thus
according to the continuity of the functional, for the point near the extremal curve
y = y(x) and the value y
near the slope function p, there is F y y (x, y, y
) = 0,
consequently J = 0. and when F y y > 0, J > 0; When F y y < 0, J < 0.
According to the definition of weak minimum (or weak maximum), the theorem is
proved. Quod erat demonstrandum.
Theorem 3.5.5 If the curve y = (x) (x 0 ≤ x ≤ x 1 ) is the extremal curve of the
functional (3.2.1) satisfying the boundary condition (3.2.2), and it satisfies the Jacobi
condition, for all points (x, y) in a zero-th order neighborhood of the extremal curve
y = (x) and any value y
, the Legendre condition holds, namely F y y (x, y, q) does
not change its sign, and the first order Taylor formula of the function F(x, y, y
) holds
at y
= p; Then the functional (3.2.1) satisfying the fixed boundary condition (3.2.2)
obtains strong extremum on the extremal curve y = (x). When F y y (x, y, q) ≥ 0, it
obtains a strong minimum; When F y y (x, y, q) ≤ 0, it obtains a strong maximum.
Proof According to the given conditions, for any y
, there is
F(x, y, y
) = F(x, y, p) + (y
− p)F p (x, y, p) +
(y
− p)
2
2!
F y y (x, y, q)
where, q is between p and y
.
Substituting the above expression into E-function, we get
E(x, y, y
, p) =
(y
− p)
2
2!
F y y (x, y, q)