218
3 Sufficient Conditions of Extrema of Functionals
If the Legendre condition holds, there should be 1 − y
2
= 1 −
y
2
1
x
2
1
> 0, namely
only when y 1 < x 1 , the extremal curve can be included in the extremal curve field
of y = cx.
3.5 Sufficient Conditions of Extrema of Functionals
After having the concepts of the above mentioned Jacobi condition, Weierstrass
condition, Legendre condition, extremal curve field, slope function and E-function
etc., the sufficient condition of extremum of the functional can be established.
3.5.1 The Weierstrass Sufficient Conditions
Theorem 3.5.1 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), the extremal curve is
included in the extremal curve field and satisfies the Jacobi condition, for all points
(x, y) close to the extremal curve y = y(x) and the value y
of the slope function
p(x, y) close to the extremal curve field, the Weierstrass condition holds, namely Efunction does not change its sign, then the functional (3.2.1) satisfying the boundary
condition (3.2.2) obtains a weak extremum on the extremal curve y = y(x). When
E ≥ 0, it obtains a weak minimum; When E ≤ 0, it obtains a weak maximum.
Proof According to the expression (3.3.10), 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
It is observed from the above expression that if E(x, y, y
, p) ≥ 0 (or ≤ 0), then
J ≥ 0 (or ≤ 0). Thus according to the definition of the weak minimum (or weak
maximum), that is proved. Quod erat demonstrandum.
Theorem 3.5.2 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), the extremal curve is
included in the extremal curve field and satisfying the Jacobi condition, for all points
(x, y) near the extremal curve y = y(x) and an arbitrary value y
, the Weierstrass
condition holds, namely E-function does not change its sign, then the functional
(3.2.1) satisfying the boundary condition (3.2.2) gets a strong extremum on the
extremal curve y = y(x). When E ≥ 0, it gets a strong minimum; When E ≤ 0, it
gets a strong maximum.
Proof According to the expression (3.3.10), the increment of the functional J [y(x)]
on the extremal curve y = (x) can be expressed as
3 Sufficient Conditions of Extrema of Functionals
If the Legendre condition holds, there should be 1 − y
2
= 1 −
y
2
1
x
2
1
> 0, namely
only when y 1 < x 1 , the extremal curve can be included in the extremal curve field
of y = cx.
3.5 Sufficient Conditions of Extrema of Functionals
After having the concepts of the above mentioned Jacobi condition, Weierstrass
condition, Legendre condition, extremal curve field, slope function and E-function
etc., the sufficient condition of extremum of the functional can be established.
3.5.1 The Weierstrass Sufficient Conditions
Theorem 3.5.1 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), the extremal curve is
included in the extremal curve field and satisfies the Jacobi condition, for all points
(x, y) close to the extremal curve y = y(x) and the value y
of the slope function
p(x, y) close to the extremal curve field, the Weierstrass condition holds, namely Efunction does not change its sign, then the functional (3.2.1) satisfying the boundary
condition (3.2.2) obtains a weak extremum on the extremal curve y = y(x). When
E ≥ 0, it obtains a weak minimum; When E ≤ 0, it obtains a weak maximum.
Proof According to the expression (3.3.10), 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
It is observed from the above expression that if E(x, y, y
, p) ≥ 0 (or ≤ 0), then
J ≥ 0 (or ≤ 0). Thus according to the definition of the weak minimum (or weak
maximum), that is proved. Quod erat demonstrandum.
Theorem 3.5.2 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), the extremal curve is
included in the extremal curve field and satisfying the Jacobi condition, for all points
(x, y) near the extremal curve y = y(x) and an arbitrary value y
, the Weierstrass
condition holds, namely E-function does not change its sign, then the functional
(3.2.1) satisfying the boundary condition (3.2.2) gets a strong extremum on the
extremal curve y = y(x). When E ≥ 0, it gets a strong minimum; When E ≤ 0, it
gets a strong maximum.
Proof According to the expression (3.3.10), the increment of the functional J [y(x)]
on the extremal curve y = (x) can be expressed as