5.3 Isoperimetric Problems
343
Solving for λ
λ = −
24J
2 a
C 2
m t
3
1
(5.6)
Substituting the expression (6) into the expression (4), we give
ω =
6a
t
3
1
t (t 1 − t)
(5.7)
Taking the derivative of ω with respect to t, we obtain
dω
dt
=
6a
t
2
1
−
12a
t
3
1
t
(5.8)
Substituting Eq. (8) into the equation of motion for the motor, the minimal
armature current I a is
I a =
1
C m
M f +
6a J
t
2
1
−
12a J
t
3
1
t
(5.9)
Thus I a is the linear function of t.
It is observed from the above discussion that the generalized isoperimetric problem
of variational methods can be transformed into the variational problem of the function
H = F + λϕ
(5.3.13)
under the integral sign. When multiplying the functions under the integral sign by
the constants, the family of extremal curves for the integral remains unchanged,
thus H can be written in the symmetrical form
H = λ 1 F + λ 2 ϕ
(5.3.14)
where, λ 1 and λ 2 both are the constants. In the expression of the function H, the
functions F and ϕ are symmetrical. This shows that, for the same physical problems,
which can be expressed by two kinds of different forms of the variational problems,
there the constraint condition in one variational problem is the variational condition in
another variational problem, if the situations of λ 1 = 0 and λ 2 = 0 is not considered,
then whether keep the integral a as a constant finding the extremum of integral J,
or keep J as a constant finding the extremum of integral a, the Euler equations are
the same, the resulting family of extremal curves are also the same. This kind of
symmetrical form is called the duality principle, principle of duality or principle
of reciprocity.
If λ 2 = 0, then H is only a constant different from F, the conditional extremal
curve of integral J will also agree with the unconditional extremal curve of this
integral, Obviously, in the general case, the extremal curve is just the conditional
343
Solving for λ
λ = −
24J
2 a
C 2
m t
3
1
(5.6)
Substituting the expression (6) into the expression (4), we give
ω =
6a
t
3
1
t (t 1 − t)
(5.7)
Taking the derivative of ω with respect to t, we obtain
dω
dt
=
6a
t
2
1
−
12a
t
3
1
t
(5.8)
Substituting Eq. (8) into the equation of motion for the motor, the minimal
armature current I a is
I a =
1
C m
M f +
6a J
t
2
1
−
12a J
t
3
1
t
(5.9)
Thus I a is the linear function of t.
It is observed from the above discussion that the generalized isoperimetric problem
of variational methods can be transformed into the variational problem of the function
H = F + λϕ
(5.3.13)
under the integral sign. When multiplying the functions under the integral sign by
the constants, the family of extremal curves for the integral remains unchanged,
thus H can be written in the symmetrical form
H = λ 1 F + λ 2 ϕ
(5.3.14)
where, λ 1 and λ 2 both are the constants. In the expression of the function H, the
functions F and ϕ are symmetrical. This shows that, for the same physical problems,
which can be expressed by two kinds of different forms of the variational problems,
there the constraint condition in one variational problem is the variational condition in
another variational problem, if the situations of λ 1 = 0 and λ 2 = 0 is not considered,
then whether keep the integral a as a constant finding the extremum of integral J,
or keep J as a constant finding the extremum of integral a, the Euler equations are
the same, the resulting family of extremal curves are also the same. This kind of
symmetrical form is called the duality principle, principle of duality or principle
of reciprocity.
If λ 2 = 0, then H is only a constant different from F, the conditional extremal
curve of integral J will also agree with the unconditional extremal curve of this
integral, Obviously, in the general case, the extremal curve is just the conditional