176
2 Variational Problems with Fixed Boundaries
the author in 2005. Theorem 2.9.2 can be called the extremal function theorem of
the complete functional. Taking after Eqs. (2.9.6)–(2.9.8), (2.9.11) can be written
as the following three forms
F u k +
S k
s k =1
(−1)
i s k D
i s k
∂ F
∂ D
i s k u k
= 0 (k = 1, 2, · · · , l)
(2.9.12)
S k
s k =0
(−1)
i s k D
i s k F D
is k u k
= 0 (k = 1, 2, · · · , l)
(2.9.13)
S k
s k =0
(−1)
i s k D
i s k
∂ F
∂ D
i s k u k
= 0 (k = 1, 2, · · · , l)
(2.9.14)
This shows that Eqs. (2.9.11)–(2.9.14) are the same as Eqs. (2.9.5)–(2.9.8) in
form, they are just one more subscript k.
Proof After the demonstration of Theorem 2.9.1, taking first variation to the functional (2.9.10), Using the property of that sequence of the derivation and variation
can be exchanged, the variations of the various terms with u k are changed into δu k ,
k = 1, 2, · · · , l, let the sum the integrals with the boundary terms be B, then there is
δ J = B +
Ω
F u 1 +
S 1
s 1 =1
(−1)
i s 1 D
i s 1 F D
is 1 u 1
δu 1 dx 1 dx 2 · · · dx m
+
Ω
F u 2 +
S 2
s 2 =1
(−1)
i s 2 D
i s 2 F D
is 2 u 2
δu 2 dx 1 dx 2 · · · dx m + · · ·
+
Ω
F u l +
S l
s l =1
(−1)
i s l D
i s l F D
is l u l
δu l dx 1 dx 2 · · · dx m = 0
(2.9.15)
where, from the necessary condition δ J = 0 of extremum of a functional, there
should be that B equals zero and the various integral term equal zero respectively,
δu 1 , δu 2 , …, δu l are arbitrary, according to the fundamental lemma of the calculus
of variation, for the various integral terms, the parts in square brackets can only be
equal to zero, thus, Eq. (2.9.11) must be obtained, and Eqs. (2.9.12)–(2.9.14) are all
the transformation of Eq. (2.9.11). Quod erat demonstrandum.
At this point, when the complete functional depending on arbitrary number of
independent variables, arbitrary number of multivariate functions and partial derivatives of arbitrary order of multivariate function obtains extremum, the conditions
which the extremal function should satisfy have been given and proved, namely any
one of Eqs. (2.9.11)–(2.9.14), these system of equations can be called the complete
system of Euler equations, united system of Euler equations or (system of) Euler
equations for short. Each complete Euler equations of Euler equations (2.9.11)–
(2.9.14) has universality, they are the united form of the various Euler equations with
2 Variational Problems with Fixed Boundaries
the author in 2005. Theorem 2.9.2 can be called the extremal function theorem of
the complete functional. Taking after Eqs. (2.9.6)–(2.9.8), (2.9.11) can be written
as the following three forms
F u k +
S k
s k =1
(−1)
i s k D
i s k
∂ F
∂ D
i s k u k
= 0 (k = 1, 2, · · · , l)
(2.9.12)
S k
s k =0
(−1)
i s k D
i s k F D
is k u k
= 0 (k = 1, 2, · · · , l)
(2.9.13)
S k
s k =0
(−1)
i s k D
i s k
∂ F
∂ D
i s k u k
= 0 (k = 1, 2, · · · , l)
(2.9.14)
This shows that Eqs. (2.9.11)–(2.9.14) are the same as Eqs. (2.9.5)–(2.9.8) in
form, they are just one more subscript k.
Proof After the demonstration of Theorem 2.9.1, taking first variation to the functional (2.9.10), Using the property of that sequence of the derivation and variation
can be exchanged, the variations of the various terms with u k are changed into δu k ,
k = 1, 2, · · · , l, let the sum the integrals with the boundary terms be B, then there is
δ J = B +
Ω
F u 1 +
S 1
s 1 =1
(−1)
i s 1 D
i s 1 F D
is 1 u 1
δu 1 dx 1 dx 2 · · · dx m
+
Ω
F u 2 +
S 2
s 2 =1
(−1)
i s 2 D
i s 2 F D
is 2 u 2
δu 2 dx 1 dx 2 · · · dx m + · · ·
+
Ω
F u l +
S l
s l =1
(−1)
i s l D
i s l F D
is l u l
δu l dx 1 dx 2 · · · dx m = 0
(2.9.15)
where, from the necessary condition δ J = 0 of extremum of a functional, there
should be that B equals zero and the various integral term equal zero respectively,
δu 1 , δu 2 , …, δu l are arbitrary, according to the fundamental lemma of the calculus
of variation, for the various integral terms, the parts in square brackets can only be
equal to zero, thus, Eq. (2.9.11) must be obtained, and Eqs. (2.9.12)–(2.9.14) are all
the transformation of Eq. (2.9.11). Quod erat demonstrandum.
At this point, when the complete functional depending on arbitrary number of
independent variables, arbitrary number of multivariate functions and partial derivatives of arbitrary order of multivariate function obtains extremum, the conditions
which the extremal function should satisfy have been given and proved, namely any
one of Eqs. (2.9.11)–(2.9.14), these system of equations can be called the complete
system of Euler equations, united system of Euler equations or (system of) Euler
equations for short. Each complete Euler equations of Euler equations (2.9.11)–
(2.9.14) has universality, they are the united form of the various Euler equations with