208
3 Sufficient Conditions of Extrema of Functionals
Thus
u 1 (x
∗
)
u 2 (x ∗ )
=
u 1 (x 0 )
u 2 (x 0 )
(3.2.13)
When endpoint B(x 1 , y 1 ) is between A and A c , there is
u(x 0 ) = η(x 0 ) = 0, u(x 1 ) = 0, η(x 1 ) = 0
This shows that u(x) and η(x) on the extremal function y = y(x) between two
points A(x 0 , y 0 ) and B(x 1 , y 1 ) can not achieve proportionality everywhere, that is to
say that η
− η
u
u
= 0, therefore there must be
η
− η
u
u
2
> 0
(3.2.14)
The Jacobi Eq. (3.2.8) can be turned into the form of second order ordinary
differential equation
η
+ p(x)η
+ q(x)η = 0
(3.2.15)
If η
= 0, then the Eq. (3.2.15) only has the unique solution of η = 0. In order to
make the Eq. (3.2.15) has a nonzero solution, let it satisfy the boundary conditions
η(x 0 ) = 0, η
(x 0 ) = 0. at the moment, any solution u(x) of the Jacobi equation
satisfying the boundary condition u(x 0 ) = 0 differs from the solution satisfying the
boundary conditions
u(x 0 ) = 0, u
(x 0 ) = 1
(3.2.16)
by only a constant factor. Thus it just needs to find the solution η = u(x) satisfying
the boundary condition (3.2.16).
Let η = u(x) be the solution that the Jacobi Eq. (3.2.8) satisfies the boundary
condition (3.2.16), if in the half-open interval [x 0 , x 1 ) u(x) has not other zero except
point x 0 , then the extremal function y = y(x) of the functional (3.2.1) is called
satisfying the Jacobi condition in the open interval (x 0 , x 1 ). Jacobi established the
condition in 1837. If in the closed interval [x 0 , x 1 ] u(x) has not other zero except
point x 0 , then the extremal function y = y(x) of the functional (3.2.1) is called
satisfying the Jacobi strong condition in the half-open interval (x 0 , x 1 ].
The solution of the Jacobi equation can be obtained from the general solution of
the Euler equation. Let y = y(x, c 1 , c 2 ) be the general solution of the Euler equation
F y −
d
dx
F y = 0
(3.2.17)
3 Sufficient Conditions of Extrema of Functionals
Thus
u 1 (x
∗
)
u 2 (x ∗ )
=
u 1 (x 0 )
u 2 (x 0 )
(3.2.13)
When endpoint B(x 1 , y 1 ) is between A and A c , there is
u(x 0 ) = η(x 0 ) = 0, u(x 1 ) = 0, η(x 1 ) = 0
This shows that u(x) and η(x) on the extremal function y = y(x) between two
points A(x 0 , y 0 ) and B(x 1 , y 1 ) can not achieve proportionality everywhere, that is to
say that η
− η
u
u
= 0, therefore there must be
η
− η
u
u
2
> 0
(3.2.14)
The Jacobi Eq. (3.2.8) can be turned into the form of second order ordinary
differential equation
η
+ p(x)η
+ q(x)η = 0
(3.2.15)
If η
= 0, then the Eq. (3.2.15) only has the unique solution of η = 0. In order to
make the Eq. (3.2.15) has a nonzero solution, let it satisfy the boundary conditions
η(x 0 ) = 0, η
(x 0 ) = 0. at the moment, any solution u(x) of the Jacobi equation
satisfying the boundary condition u(x 0 ) = 0 differs from the solution satisfying the
boundary conditions
u(x 0 ) = 0, u
(x 0 ) = 1
(3.2.16)
by only a constant factor. Thus it just needs to find the solution η = u(x) satisfying
the boundary condition (3.2.16).
Let η = u(x) be the solution that the Jacobi Eq. (3.2.8) satisfies the boundary
condition (3.2.16), if in the half-open interval [x 0 , x 1 ) u(x) has not other zero except
point x 0 , then the extremal function y = y(x) of the functional (3.2.1) is called
satisfying the Jacobi condition in the open interval (x 0 , x 1 ). Jacobi established the
condition in 1837. If in the closed interval [x 0 , x 1 ] u(x) has not other zero except
point x 0 , then the extremal function y = y(x) of the functional (3.2.1) is called
satisfying the Jacobi strong condition in the half-open interval (x 0 , x 1 ].
The solution of the Jacobi equation can be obtained from the general solution of
the Euler equation. Let y = y(x, c 1 , c 2 ) be the general solution of the Euler equation
F y −
d
dx
F y = 0
(3.2.17)