Chapter 5
Variational Problems of Conditional
Extrema
The variational problems encountered in the natural sciences and engineering technology, sometimes require that the extremal function in addition to satisfying the
given boundary conditions, also satisfies certain additional conditions, this is the
condition extremal problems for a functional. The extremum obtained by a functional in satisfying some additional conditions is called the conditional extremum.
Attaching some constraint conditions to the function that a functional depends on
to find the extremal problems of the functional is called the variational problem
of conditional extremum. The conditional extremum of functional with holonomic
constraint, differential constraint and isoperimetric problem will be discussed in this
chapter, and the extremal problems the simple mixed type functional are discussed.
The computational method of conditional extremum for a functional is similar
to the computational method of conditional extremum for a function, which can be
realized with Lagrange multiplier method, that is to choose a new functional, such
that the conditional extremal problem of the original functional is transformed into
an equivalent unconditional extremal problem.
5.1 Variational Problems with Holonomic Constraints
This section mainly studies the extremal problem of the functional under the
constraint condition and boundary condition, and derives the condition which the
extremum of the functional should satisfy.
Let the functional
J [y] =
x 1
x 0
F(x, y 1 , y 2 , . . . , y n , y
1 , y
2 , . . . , y
n )dx
(5.1.1)
The constraint conditions are
© Beijing Institute of Technology Press and Springer Nature Singapore Pte Ltd. 2021
D. Lao and S. Zhao, Fundamental Theories and Their Applications of the Calculus
of Variations, https://doi.org/10.1007/978-981-15-6070-5_5
319
Variational Problems of Conditional
Extrema
The variational problems encountered in the natural sciences and engineering technology, sometimes require that the extremal function in addition to satisfying the
given boundary conditions, also satisfies certain additional conditions, this is the
condition extremal problems for a functional. The extremum obtained by a functional in satisfying some additional conditions is called the conditional extremum.
Attaching some constraint conditions to the function that a functional depends on
to find the extremal problems of the functional is called the variational problem
of conditional extremum. The conditional extremum of functional with holonomic
constraint, differential constraint and isoperimetric problem will be discussed in this
chapter, and the extremal problems the simple mixed type functional are discussed.
The computational method of conditional extremum for a functional is similar
to the computational method of conditional extremum for a function, which can be
realized with Lagrange multiplier method, that is to choose a new functional, such
that the conditional extremal problem of the original functional is transformed into
an equivalent unconditional extremal problem.
5.1 Variational Problems with Holonomic Constraints
This section mainly studies the extremal problem of the functional under the
constraint condition and boundary condition, and derives the condition which the
extremum of the functional should satisfy.
Let the functional
J [y] =
x 1
x 0
F(x, y 1 , y 2 , . . . , y n , y
1 , y
2 , . . . , y
n )dx
(5.1.1)
The constraint conditions are
© Beijing Institute of Technology Press and Springer Nature Singapore Pte Ltd. 2021
D. Lao and S. Zhao, Fundamental Theories and Their Applications of the Calculus
of Variations, https://doi.org/10.1007/978-981-15-6070-5_5
319