2.8 Variational Problems Depending on Functions of Several Variables
159
2.8 Variational Problems Depending on Functions
of Several Variables
In many engineering and physical problems, the extremal problem of the functional
depending on the multivariate function would be often met, such as plane problem in
elastic mechanics, the plane electric field problem in electromagnetism, both contain
two independent variables x, y. The vibration problem of the plate in elastic dynamics
contains three independent variables x, y, t. The unsteady heat conduction equation
in heat transfer contains four independent variables x, y, z and t. The variational
problem of the functional with two independent variables is discussed emphatically
in this section.
Theorem 2.8.1 Let D be a plane domain, (x, y) ∈ D, u(x, y) ∈ C
2
(D), the
functional
J [u(x, y)] =
¨
D
F(x, y, u, u x , u y )dxdy
(2.8.1)
obtain extremum, then the extremal function u = u(x, y) which has a known value
on the boundary L of the domain D must satisfy the partial differential equation
F u −
∂
∂ x
F u x −
∂
∂ y
F u y = 0
(2.8.2)
The equation is called the Ostrogradsky equation. This equation is obtained in
1834 by Russian mathematician Ostrogradsky first. The Ostrogradsky equation is
the further development of the Euler equation, sometimes it is also called the Euler
equation.
∂
∂ x
F u x and
∂
∂ y
F u y in Eq. (2.8.2) are the completely partial derivative with respect
to the independent variables x and y, they are respectively
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
∂
∂ x
F u x = F u x x + F u x u u x + F u x u x u xx + F u x u y u yx
∂
∂ y
F u y = F u y y + F u y u u y + F u y u x u xy + F u y u y u yy
(2.8.3)
Substituting Eq. (2.8.3) into Eq. (2.8.2), we obtain the expansion of Eq. (2.8.2)
F u x u x u xx + 2F u x u y u xy + F u y u y u yy + F u x u u x + F u y u u y + F xu x + F yu y − F u = 0
(2.8.4)
Equation (2.8.4) is the second order partial differential equation, the boundary
condition is that u is a known value on the boundary L of D.
Proof Let u = u(x, y) be the extremal curve of the functional J [u(x, y)], making
the nearby curve of u = u(x, y)
159
2.8 Variational Problems Depending on Functions
of Several Variables
In many engineering and physical problems, the extremal problem of the functional
depending on the multivariate function would be often met, such as plane problem in
elastic mechanics, the plane electric field problem in electromagnetism, both contain
two independent variables x, y. The vibration problem of the plate in elastic dynamics
contains three independent variables x, y, t. The unsteady heat conduction equation
in heat transfer contains four independent variables x, y, z and t. The variational
problem of the functional with two independent variables is discussed emphatically
in this section.
Theorem 2.8.1 Let D be a plane domain, (x, y) ∈ D, u(x, y) ∈ C
2
(D), the
functional
J [u(x, y)] =
¨
D
F(x, y, u, u x , u y )dxdy
(2.8.1)
obtain extremum, then the extremal function u = u(x, y) which has a known value
on the boundary L of the domain D must satisfy the partial differential equation
F u −
∂
∂ x
F u x −
∂
∂ y
F u y = 0
(2.8.2)
The equation is called the Ostrogradsky equation. This equation is obtained in
1834 by Russian mathematician Ostrogradsky first. The Ostrogradsky equation is
the further development of the Euler equation, sometimes it is also called the Euler
equation.
∂
∂ x
F u x and
∂
∂ y
F u y in Eq. (2.8.2) are the completely partial derivative with respect
to the independent variables x and y, they are respectively
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
∂
∂ x
F u x = F u x x + F u x u u x + F u x u x u xx + F u x u y u yx
∂
∂ y
F u y = F u y y + F u y u u y + F u y u x u xy + F u y u y u yy
(2.8.3)
Substituting Eq. (2.8.3) into Eq. (2.8.2), we obtain the expansion of Eq. (2.8.2)
F u x u x u xx + 2F u x u y u xy + F u y u y u yy + F u x u u x + F u y u u y + F xu x + F yu y − F u = 0
(2.8.4)
Equation (2.8.4) is the second order partial differential equation, the boundary
condition is that u is a known value on the boundary L of D.
Proof Let u = u(x, y) be the extremal curve of the functional J [u(x, y)], making
the nearby curve of u = u(x, y)