Chapter 4
Problems with Variable Boundaries
In the foregoing study of extremal problem of a functional, assume the limits of integral are fixed, namely the admissible functions all pass through two fixed endpoints A
and B. But in many practical problems, the limit of integral for a functional can either
be fixed, or be undetermined. If one or two endpoints of the admissible function does
not pass through a given point in advance, but having to go through variation can be
determined, then the endpoint is called the variable endpoint or mobile endpoint.
For a function of one variable, the endpoint and boundary have the same meaning,
hence the above mentioned endpoint can also be called the variable boundary or
moving boundary, sometimes it is called the undetermined boundary or undetermined endpoint. The two endpoints of admissible functions are the upper limit and
lower limit of integral for a functional. If the limit of integral for a functional is variable, or the domain of integral is given but lack of boundary conditions, then such
the variational problem is called the variational problem of variable boundary
or variational problem of undetermined boundary. When the boundary value of
the admissible functions of a functional is not explicitly given, such the variational
problem is called the unconstrained variational problem. This chapter will discuss
the above mentioned variational problems.
4.1 Variational Problems of the Simplest Functional
Let the functional
J [y(x)] =
x 1
x 0
F(x, y, y
)dx
(4.1.1)
the admissible curve y = y(x) ∈ C
2 class function, and two endpoints A(x 0 , y 0 ),
B(x 1 , y 1 ) move on y = ϕ(x) and y = ψ(x) of the two given C
2 class function
© 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_4
239
Problems with Variable Boundaries
In the foregoing study of extremal problem of a functional, assume the limits of integral are fixed, namely the admissible functions all pass through two fixed endpoints A
and B. But in many practical problems, the limit of integral for a functional can either
be fixed, or be undetermined. If one or two endpoints of the admissible function does
not pass through a given point in advance, but having to go through variation can be
determined, then the endpoint is called the variable endpoint or mobile endpoint.
For a function of one variable, the endpoint and boundary have the same meaning,
hence the above mentioned endpoint can also be called the variable boundary or
moving boundary, sometimes it is called the undetermined boundary or undetermined endpoint. The two endpoints of admissible functions are the upper limit and
lower limit of integral for a functional. If the limit of integral for a functional is variable, or the domain of integral is given but lack of boundary conditions, then such
the variational problem is called the variational problem of variable boundary
or variational problem of undetermined boundary. When the boundary value of
the admissible functions of a functional is not explicitly given, such the variational
problem is called the unconstrained variational problem. This chapter will discuss
the above mentioned variational problems.
4.1 Variational Problems of the Simplest Functional
Let the functional
J [y(x)] =
x 1
x 0
F(x, y, y
)dx
(4.1.1)
the admissible curve y = y(x) ∈ C
2 class function, and two endpoints A(x 0 , y 0 ),
B(x 1 , y 1 ) move on y = ϕ(x) and y = ψ(x) of the two given C
2 class function
© 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_4
239