244
4 Problems with Variable Boundaries
Substituting the expression (4.1.16) into the expression (4.1.12), when δx 1 , δy
and δy 1 are very small, neglecting the high order small quantity F y
x=x 1
εδx 1 and
εδx 1 , then the main linear part of J with respect to δx 1 , δy and δy 1 namely the first
variation can be obtained
δ J =
x 1
x 0
F y −
d
dx
F y
δydx + F| x=x 1 δx 1 + F y
x=x 1
[δy 1 − y
(x 1 )δx 1 ]
=
x 1
x 0
F y −
d
dx
F y
δydx + (F − y
F y )
x=x 1
δx 1 + F y
x=x 1
δy 1 (4.1.20)
Because the extremum of a functional only can be obtained on the extremal curve,
therefore F y −
d
dx
F y ≡ 0. Thus the expression (4.1.20) can be written as
δ J = F| x=x 1 δx 1 + F y
x=x 1
[δy 1 − y
(x 1 )δx 1 ]
= (F − y
F y )
x=x 1
δx 1 + F y
x=x 1
δy 1
(4.1.21)
Then by the condition δ J = 0, we give
(F − y
F y )
x=x 1
δx 1 + F y
x=x 1
δy 1 = 0
(4.1.22)
If δx 1 has nothing to do with δy 1 each other, then by the expression (4.1.22), we
get
(F − y
F y )
x=x 1
= 0
(4.1.23)
F y
x=x 1
= 0
(4.1.24)
However, it is necessary to consider the case of δx 1 related to δy 1 , it is that
when substituting Eq. (4.1.24) into Eq. (4.1.23), F| x=x 1 = 0 can be obtained, this
means that the integrand of the functional is zero, the situation is just a special case
of variational problem, in general it is not true, it need not be thought about. Of
cause the functional of F| x=x 1 = 0 also can be found, for instance the functional
J [y] =
x 1
x 0
y
2 dx, its minimum is zero, the integrand of the functional also is zero.
When point B moves along the straight line x = x 1 , then δx 1 = 0, while δy 1 is
arbitrary, at this time there is only Eq. (4.1.24) and no Eq. (4.1.23).
Equation (4.1.24) is the condition satisfied by free endpoint B, it is the boundary
condition derived based on the necessary condition δ J = 0 of extremum of a functional, this boundary condition is called the natural boundary condition or kinetic
boundary condition, meanwhile the fixed boundary condition is called the forcing
boundary condition or essential boundary condition or geometric boundary
4 Problems with Variable Boundaries
Substituting the expression (4.1.16) into the expression (4.1.12), when δx 1 , δy
and δy 1 are very small, neglecting the high order small quantity F y
x=x 1
εδx 1 and
εδx 1 , then the main linear part of J with respect to δx 1 , δy and δy 1 namely the first
variation can be obtained
δ J =
x 1
x 0
F y −
d
dx
F y
δydx + F| x=x 1 δx 1 + F y
x=x 1
[δy 1 − y
(x 1 )δx 1 ]
=
x 1
x 0
F y −
d
dx
F y
δydx + (F − y
F y )
x=x 1
δx 1 + F y
x=x 1
δy 1 (4.1.20)
Because the extremum of a functional only can be obtained on the extremal curve,
therefore F y −
d
dx
F y ≡ 0. Thus the expression (4.1.20) can be written as
δ J = F| x=x 1 δx 1 + F y
x=x 1
[δy 1 − y
(x 1 )δx 1 ]
= (F − y
F y )
x=x 1
δx 1 + F y
x=x 1
δy 1
(4.1.21)
Then by the condition δ J = 0, we give
(F − y
F y )
x=x 1
δx 1 + F y
x=x 1
δy 1 = 0
(4.1.22)
If δx 1 has nothing to do with δy 1 each other, then by the expression (4.1.22), we
get
(F − y
F y )
x=x 1
= 0
(4.1.23)
F y
x=x 1
= 0
(4.1.24)
However, it is necessary to consider the case of δx 1 related to δy 1 , it is that
when substituting Eq. (4.1.24) into Eq. (4.1.23), F| x=x 1 = 0 can be obtained, this
means that the integrand of the functional is zero, the situation is just a special case
of variational problem, in general it is not true, it need not be thought about. Of
cause the functional of F| x=x 1 = 0 also can be found, for instance the functional
J [y] =
x 1
x 0
y
2 dx, its minimum is zero, the integrand of the functional also is zero.
When point B moves along the straight line x = x 1 , then δx 1 = 0, while δy 1 is
arbitrary, at this time there is only Eq. (4.1.24) and no Eq. (4.1.23).
Equation (4.1.24) is the condition satisfied by free endpoint B, it is the boundary
condition derived based on the necessary condition δ J = 0 of extremum of a functional, this boundary condition is called the natural boundary condition or kinetic
boundary condition, meanwhile the fixed boundary condition is called the forcing
boundary condition or essential boundary condition or geometric boundary