112
2 Variational Problems with Fixed Boundaries
Theorem 2.3.1 If the functional J [y(x)] attains an extremum on y = y(x), then its
variation δ J on y = y(x) is equal to zero.
Proof For y = y(x) and arbitrary fixed δy, J [y(x) + εδy] = Φ(ε) is the function of
the variable ε, when ε = 0, the function Φ(ε) obtains extremum, hence Φ
(0) = 0,
but according to Eqs. (2.3.15) and (2.3.32), it follows that the variation of J [y(x)]
on y = y(x) is δ J = 0. Quod erat demonstrandum.
The variation δ J = 0 of the functional J [y] is called the necessary condition or
condition of stationary value of a functional obtaining extremum, it is also called
the variational equation of the functional J [y]. Sometimes the variation δ J = 0 of
the functional J [y] is called the variational principle.
In the first variation expression (2.3.15), the integrals under the integral sign are
the linear functions of δy and δy
. Using integration by parts, the variations can be
changed into that the integrals under integral sign are just the linear functions of
δy, such the transformation is called the Lagrange transformation; Or it is just the
linear functions of δy
, this transformation is called the Riemann transformation.
The Lagrange transformation is as follows:
Using integration by parts to the second term under integral sign in Eq. (2.3.15),
we obtain
x 1
x 0
F y δy
dx = F y δy
x 1
x 0
−
x 1
x 0
δy
d
dx
F y dx
(2.3.34)
If the variation is equal to zero at points x 0 and x 1 , then
x 1
x 0
F y δy
dx = −
x 1
x 0
δy
d
dx
F y dx
(2.3.35)
Thus there is
δ J =
x 1
x 0
(F y δy + F y δy
)dx =
x 1
x 0
F y −
d
dx
F y
δydx
(2.3.36)
Note that in the above definition of variational, once assuming that the function
y = y(x) is differentiable, but it is not assumed that y
is also differentiable, therefore
the Lagrange transformation is illegal.
In order to eliminate the additional assumptions that the second derivative y
exists, Riemann put forward another variational transform, that is expressed with
notation
x
x 0
F y dx = N (x)
(2.3.37)
Thus there is
2 Variational Problems with Fixed Boundaries
Theorem 2.3.1 If the functional J [y(x)] attains an extremum on y = y(x), then its
variation δ J on y = y(x) is equal to zero.
Proof For y = y(x) and arbitrary fixed δy, J [y(x) + εδy] = Φ(ε) is the function of
the variable ε, when ε = 0, the function Φ(ε) obtains extremum, hence Φ
(0) = 0,
but according to Eqs. (2.3.15) and (2.3.32), it follows that the variation of J [y(x)]
on y = y(x) is δ J = 0. Quod erat demonstrandum.
The variation δ J = 0 of the functional J [y] is called the necessary condition or
condition of stationary value of a functional obtaining extremum, it is also called
the variational equation of the functional J [y]. Sometimes the variation δ J = 0 of
the functional J [y] is called the variational principle.
In the first variation expression (2.3.15), the integrals under the integral sign are
the linear functions of δy and δy
. Using integration by parts, the variations can be
changed into that the integrals under integral sign are just the linear functions of
δy, such the transformation is called the Lagrange transformation; Or it is just the
linear functions of δy
, this transformation is called the Riemann transformation.
The Lagrange transformation is as follows:
Using integration by parts to the second term under integral sign in Eq. (2.3.15),
we obtain
x 1
x 0
F y δy
dx = F y δy
x 1
x 0
−
x 1
x 0
δy
d
dx
F y dx
(2.3.34)
If the variation is equal to zero at points x 0 and x 1 , then
x 1
x 0
F y δy
dx = −
x 1
x 0
δy
d
dx
F y dx
(2.3.35)
Thus there is
δ J =
x 1
x 0
(F y δy + F y δy
)dx =
x 1
x 0
F y −
d
dx
F y
δydx
(2.3.36)
Note that in the above definition of variational, once assuming that the function
y = y(x) is differentiable, but it is not assumed that y
is also differentiable, therefore
the Lagrange transformation is illegal.
In order to eliminate the additional assumptions that the second derivative y
exists, Riemann put forward another variational transform, that is expressed with
notation
x
x 0
F y dx = N (x)
(2.3.37)
Thus there is