5.4 Extremal Problems of Mixed Type Functionals
345
The exremal problem of the mixed type functional is called the mixed variational
problem or generalized variational problem.
The structure of the mixed type functional can be quite complicated, it is difficult
to make general argument, only the variational problems of the simplest type of
the mixed type functional, two-dimensional, three-dimensional and n-dimensional
functional are discussed in this section.
5.4.1 Extremal Problems of Simple Mixed Type Functionals
Considering the following functional
J =
x 1
x 0
F(x, y, y
)dx + Φ(x 0 , y 0 , x 1 , y 1 )
(5.4.1)
where, the coordinates (x 0 , y 0 ), (x 1 , y 1 ) of the variable boundary points can
be constained by some conditions, for instance y 0 = ϕ(x 0 ), y 1 = ψ(x 1 );
Φ(x 0 , y 0 , x 1 , y 1 ) is a constant term. The variational problem of the mixed type functional (5.4.1) is also called the Bolza(’s) problem. In particular, when Φ ≡ 0, it
is called the Lagrange problem. When F ≡ 0, it is called the Mayer problem.
It is thus clear that, the Bolza problem has the most general form, the Lagrange
problem and Mayer problem both are the particular cases of the Bolza problem.
If some auxiliary variables are introduced, these three problems can be mutually
transformed. For example, if let F = H −
dΦ
dx
, then the Bolza problem can be transformed to equivalent Lagrange problem. If the function G(x, y) is introduced, such
that
d
dx
G(x, y) = F(x, y, y
), then the Bolza problem can also be transformed to
equivalent Mayer problem.
Obviously, the extremum of the functional (5.4.1) can only achieve on the solution
of the Euler equation
F y −
d
dx
F y = 0
(5.4.2)
According to the expression (4.1.21), there is
δ J = [(F − y
F y )δx 1 + F y δy 1 ]
x=x 1
− [(F − y
F y )δx 0 + F y δy 0 ]
x=x 0
+
Φ x 0 δx 0 + Φ y 0 δy 0 + Φ x 1 δx 1 + Φ y 1 δy 1 =
(F − y
F y + Φ x 1 )
x=x 1
δx 1 + (F y + Φ y 1 )
x=x 1
δy 1 −
(F − y
F y − Φ x 0 )
x=x 0
δx 0 − (F y − Φ y 0 )
x=x 0
δy 0
(5.4.3)
When x 0 and x 1 are equal to constants respectively, then the endpoints can only
variate upward or downward, thus the natural boundary conditions are obtained. At
345
The exremal problem of the mixed type functional is called the mixed variational
problem or generalized variational problem.
The structure of the mixed type functional can be quite complicated, it is difficult
to make general argument, only the variational problems of the simplest type of
the mixed type functional, two-dimensional, three-dimensional and n-dimensional
functional are discussed in this section.
5.4.1 Extremal Problems of Simple Mixed Type Functionals
Considering the following functional
J =
x 1
x 0
F(x, y, y
)dx + Φ(x 0 , y 0 , x 1 , y 1 )
(5.4.1)
where, the coordinates (x 0 , y 0 ), (x 1 , y 1 ) of the variable boundary points can
be constained by some conditions, for instance y 0 = ϕ(x 0 ), y 1 = ψ(x 1 );
Φ(x 0 , y 0 , x 1 , y 1 ) is a constant term. The variational problem of the mixed type functional (5.4.1) is also called the Bolza(’s) problem. In particular, when Φ ≡ 0, it
is called the Lagrange problem. When F ≡ 0, it is called the Mayer problem.
It is thus clear that, the Bolza problem has the most general form, the Lagrange
problem and Mayer problem both are the particular cases of the Bolza problem.
If some auxiliary variables are introduced, these three problems can be mutually
transformed. For example, if let F = H −
dΦ
dx
, then the Bolza problem can be transformed to equivalent Lagrange problem. If the function G(x, y) is introduced, such
that
d
dx
G(x, y) = F(x, y, y
), then the Bolza problem can also be transformed to
equivalent Mayer problem.
Obviously, the extremum of the functional (5.4.1) can only achieve on the solution
of the Euler equation
F y −
d
dx
F y = 0
(5.4.2)
According to the expression (4.1.21), there is
δ J = [(F − y
F y )δx 1 + F y δy 1 ]
x=x 1
− [(F − y
F y )δx 0 + F y δy 0 ]
x=x 0
+
Φ x 0 δx 0 + Φ y 0 δy 0 + Φ x 1 δx 1 + Φ y 1 δy 1 =
(F − y
F y + Φ x 1 )
x=x 1
δx 1 + (F y + Φ y 1 )
x=x 1
δy 1 −
(F − y
F y − Φ x 0 )
x=x 0
δx 0 − (F y − Φ y 0 )
x=x 0
δy 0
(5.4.3)
When x 0 and x 1 are equal to constants respectively, then the endpoints can only
variate upward or downward, thus the natural boundary conditions are obtained. At