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The secret of the calculus of variations: How to find the shortest path through small changes?
In the world of mathematical analysis, the calculus of variations is an important tool for exploring extreme value problems. This field explores how to find the maximum or minimum of a function or functional through small changes. Functional functions can be understood as a way to map a set of functions to real numbers, and the core of the variational method is to analyze how these mappings are affected by small changes. This article will delve into the history, basic concepts and applications of the calculus of variations, especially the mystery of how to find the shortest path.
The calculus of variations allows us to explore extreme values, find the best path from one point to another, and can even be applied to the principle of least action in physics.
History of calculus of variations
The origins of the calculus of variations date back to the 17th century, when Newton posed the problem of least resistance. Later, Johann Bernoulli introduced the famous "steepest descent line problem" in 1696. Since then, this field has aroused strong interest among mathematicians. Among them, Leonhard Euler was the first scholar to elaborate on the calculus of variations in depth and published his research results in 1733. His work influenced subsequent mathematicians such as Lagrange and Legendre, who further expanded the theory of calculus of variations.
Basic concepts of calculus of variations
The purpose of the calculus of variations is to find extreme values, which are usually the maximum or minimum of a function. The extreme value of a functional is called an extremal function. If a functional reaches a local minimum at a certain function, the function is the so-called extremal function.
In the calculus of variations, the most well-known equation is the Euler-Lagrange equation, which is an important tool for finding extreme functions.
Extremal values and the Euler-Lagrange equation
Imagine a functional corresponding to the length of a curve. The variational method analyzes small changes in the curve to find the shortest path. When the two endpoints of a curve are given, without any restrictions, the simplest solution is a straight line. However, for some constraints, the optimal solution may no longer be a straight line, but a complex curve living in two or three dimensions.
The variational method is not only applicable to mathematical problems, but also to natural phenomena. For example, when light passes through a medium, it follows the principle of the shortest optical path.
Calculus of variations applied to physics
In physics, the method of variation is widely used, especially in mechanics, where the principle of least action is one of its applications. This principle states that an object will move along a path that minimizes the amount of action during motion. This concept reveals the close connection between calculus of variations and physical phenomena, demonstrating the interactive influence of mathematics and natural science.
Plane problems and minimal surfaces
The calculus of variations also provides solutions when dealing with minimal surface problems, such as Plato's problem. The Plato problem requires finding a surface with the smallest area that must cover a given contour. Through simple experiments, we can find that the bubble formed by a frame soaked in soapy water is the smallest surface that satisfies this condition.
However, although these experiments are relatively easy to operate, the mathematical description behind them is quite complicated, and there are multiple local minimum solutions.
Evolving theories and tools
Over time, the theory of calculus of variations gradually matured and attracted more and more mathematicians to participate in research. From Karl Weierstrass in the 19th century to Amy Noether in the 20th century, the contribution of each mathematician improved the theory of calculus of variations. Especially in the development of optimal control theory and dynamic programming, the variation method once again showed its importance.
Conclusion
The calculus of variations provides a powerful tool for exploring and solving complex optimization problems. Whether in mathematics, physics, or engineering, applications of the calculus of variations are endless and continue to evolve as new technologies become available. Facing the future, how will deeper applications of the calculus of variations change the way we solve problems?
