3.2 The Jacobi Conditions and Jacobi Equation
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where
J 2 =
x 1
x 0
(F yy η
2
+ 2F yy ηη
+ F y y η
2
)dx
(3.2.5)
According to Eq. (3.2.4), the second variation δ
2 J and J 2 have the same sign.
Since y(x) and y
(x) on which F yy , F yy and F y y depending in Eq. (3.2.4) can
all be obtained by solving the Euler equation, they are the known function of x
after substituting the extremal function, so J 2 can be regarded as the functional only
depending on η. Thus there is the following preparation theorem:
Preparation Theorem 3.2.1 If the functional (3.2.5) obtains the extremum η = u(x),
then
F yy −
d
dx
F yy
u −
d
dx
F y y
du
dx
= 0
(3.2.6)
Equation (3.2.6) is called the Jacobi accessory equation or Jacobi equation.
If F y y = 0, then the Jacobi equation is the second order linear homogeneous
differential equation about u. If let
S = F yy −
d
dx
F yy , R = F y y
(3.2.7)
then the Jacobi Eq. (3.2.6) can be abbreviated to
Su −
d
dx
(Ru
) = 0
(3.2.8)
Proof Let G = G(x, η, η
) = F yy η
2
+ 2F yy ηη
+ F y y η
2 , then the Euler equation
of J 2 is
G η −
d
dx
G η = 0
namely
2F yy η + 2F yy η
−
d
dx
(2F yy η + 2F y y η
) = 0
(3.2.9)
Deriving the first term in brackets of the above equation, eliminating common
factor 2, and taking note that η = u(x), the Eq. (3.2.6) can be obtained. Quod erat
demonstrandum.