Chapter 6
Variational Problems in Parametric
Forms
The extremal curves of the functionals researched in the previous chapters are all
expressed in the forms of the explicit functions, each extremal curve and the straight
line parallel to the Oy axis can only have one intersection point, this restriction
reduces the researched range, If representing the curve in the parameter form, which
can get rid of this restriction, provide convenient for the research of multivalued
function. The variational problems of the parameter form with the fixed boundary,
variable boundary and the realization conditions are discussed in this chapter.
6.1 Parametric Forms of Curves and Homogeneous
Condition
Considering the isoperimetric problem, let the length of the closed curve be L, the
parametric equations are
x = x(t), y = y(t) (t 0 t t 1 )
(6.1.1)
where, the function x(t), y(t) continuously differential, with x(t 0 ) = x(t 1 ), y(t 0 ) =
y(t 1 ).
The area surrounded by the curve is
A =
1
2
L
(xdy − ydx) =
1
2
L
(x ˙
y − y ˙
x)dt
(6.1.2)
The length of the curve can be expressed as
L =
t 1
t 0
˙
x 2 (t) + ˙
y 2 (t)dt
(6.1.3)
© 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_6
365
Variational Problems in Parametric
Forms
The extremal curves of the functionals researched in the previous chapters are all
expressed in the forms of the explicit functions, each extremal curve and the straight
line parallel to the Oy axis can only have one intersection point, this restriction
reduces the researched range, If representing the curve in the parameter form, which
can get rid of this restriction, provide convenient for the research of multivalued
function. The variational problems of the parameter form with the fixed boundary,
variable boundary and the realization conditions are discussed in this chapter.
6.1 Parametric Forms of Curves and Homogeneous
Condition
Considering the isoperimetric problem, let the length of the closed curve be L, the
parametric equations are
x = x(t), y = y(t) (t 0 t t 1 )
(6.1.1)
where, the function x(t), y(t) continuously differential, with x(t 0 ) = x(t 1 ), y(t 0 ) =
y(t 1 ).
The area surrounded by the curve is
A =
1
2
L
(xdy − ydx) =
1
2
L
(x ˙
y − y ˙
x)dt
(6.1.2)
The length of the curve can be expressed as
L =
t 1
t 0
˙
x 2 (t) + ˙
y 2 (t)dt
(6.1.3)
© 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_6
365