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The Minimal Surface Problem: How do planar boundaries give rise to fascinating three-dimensional shapes?
In the field of mathematical analysis, "variational method" is a crucial branch that focuses on finding the extreme values of function mappings, which are called "functionals". The study of functionals often involves defining integrals that cover functions and their derivatives, which makes the calculus of variations a powerful tool for finding extreme values. One of the most common examples is finding the shortest curve between two points, which, if unconstrained, would be the straight line between the two points. However, when the curve is constrained to a three-dimensional surface, the solution is no longer obvious, leading to a series of fascinating mathematical problems.
In the absence of constraints, the shortest path is a straight line, but in a restricted environment, the complexity of the solution increases, and there may even be multiple possible solutions.
The application of the calculus of variations is not limited to the shortest distance problem. For example, according to Fermat's principle, the path of light follows the principle of shortest optical path, which is closely related to the properties of the medium. From a mechanical point of view, this principle can also be compared to the principle of minimum action. Many important problems involve functions of many variables, such as the boundary value problem of Laplace's equations, which satisfies the Derek-Ley principle. When dealing with minimum surface problems on planar boundaries, it is a matter of finding the minimum area, which can be intuitively experimented with by dipping the frame in soapy water.
Mathematically, although these experiments are relatively easy to perform, the mathematics behind them is far from simple, because there may be more than one local minimum surface, and these surfaces may have nontrivial topological shapes.
Historical Background of the Calculus of Variations
The history of the calculus of variations dates back to the late 17th century, when Newton's problem of least resistance was first proposed in 1687, followed by the shortest path problem proposed by John Barnary in 1696, which quickly attracted Jacob Barnary. The attention of Nari and Marquis Rahport and others. The calculus of variations began to acquire a formalized status with the further development of the subject by Leonhard Euler in 1733. Next, Joseph Louis Lagrange, inspired by Euler's work, made important contributions to the theory.Lagrange's work turned the calculus of variations into a purely analytical method, and it was formally named the calculus of variations in his 1756 speech.
With the progress of the times, mathematicians such as Adrien-Marie Legendre, Carl Friedrich Gauss, Simeon Poisson and others have made numerous contributions to this field. contribute. The work of Karl Wilstrasse is considered the most important achievement of the century, placing the theory of the calculus of variations on a solid foundation. The 20th century was another heyday for the calculus of variations, with mathematicians such as David Hilbert and Emmy Noether further advancing the theory.
Extrema of the calculus of variations and the Euler-Lagrange equation
The core of the calculus of variations is to find the maximum or minimum value of the functional, which are collectively referred to as "extreme values". A functional maps a function space to a scalar, which allows functionals to be described as "functions of functions". To find the extrema of a functional, we often use the Euler-Lagrange equations. The basic idea of this equation is similar to the way we find extrema of a function by looking for its derivative to be zero, but in the case of functionals, we look for functions that make the derivative of the functional zero.
By solving the Euler-Lagrange equations, we can find the extrema of the functional, which provides the structure for the calculus of variations.
Whether in physics, engineering, or other areas of mathematics, the calculus of variations has demonstrated its power and flexibility. In many applications, whether in the shortest path or minimum surface problem, the calculus of variations has been shown to generate a wide variety of solutions. These solutions are often not just simple geometric shapes; they may contain deeper mathematical meanings and may be able to explain many natural phenomena.
With the progress of mathematics, our understanding of the calculus of variations is becoming deeper and broader. In the future, how will it further guide us to explore unknown mathematical and physical problems?
