中文版

什么是变分法（Calculus of Variations）？

变分法（Calculus of Variations）是一种数学方法，用于求解在某种条件下，使某个函数达到极值（最大值或最小值）的变量。变分法主要用于优化问题，尤其是在物理、工程、经济学等领域。通过变分法，我们能够解决一些看似复杂的最优化问题，找到最优解。

简单来说，变分法的目标是寻找一个函数，使得它在某个定义的函数空间中，满足特定的条件下，某个量（通常是积分形式的目标函数）取得极值。

变分法的基本思想

变分法的基本思想是：给定一个目标函数 ( $J[y]$ )，它是某个函数 ( $y(x)$ ) 在某一区间 ( $[a,b]$ ) 上的积分。我们希望找到一个函数 ( $y(x)$ )，使得这个目标函数 ( $J[y]$ ) 取得极值。

我们将目标函数 ( $J[y]$ ) 定义为：
$$J[y]={\int }_a^{b}F(x,y(x),y^{\prime }(x))dx$$
其中，( $F(x,y(x),y^{\prime }(x))$ ) 是关于 ( $x$ )、( $y(x)$ ) 以及 ( $y^{\prime }(x)$ ) 的函数，通常被称为拉格朗日函数（Lagrangian）。我们的目标是通过变分法来求解使得 ( $J[y]$ ) 极值的函数 ( $y(x)$ )。

欧拉-拉格朗日方程

通过变分法求解极值问题时，我们需要对目标函数进行变分，从而得到欧拉-拉格朗日方程，这是变分法的核心。

欧拉-拉格朗日方程的推导基于一个重要的概念——变分。假设 ( $y(x)$ ) 是我们需要求解的最优函数，并且我们引入一个小的变化 ( $\eta (x)$ )，使得 ( $y(x)$ ) 变成 ( $y(x)+ϵ\eta (x)$ )，其中 ( $ϵ$ ) 是一个小参数。我们考察在这种变化下，目标函数的变动情况。

将目标函数变为：
$$J[y+ϵ\eta ]={\int }_a^{b}F(x,y(x)+ϵ\eta (x),y^{\prime }(x)+ϵ{\eta }^{\prime }(x))dx$$
为了使 ( $J[y]$ ) 取得极值，我们需要对 ( $ϵ$ ) 求导并令其为零，得到以下方程：
$$\frac{d}{dϵ}J[y+ϵ\eta ]{|}_{ϵ=0}=0$$
这就得到欧拉-拉格朗日方程：
$$\frac{\partial F}{\partial y}-\frac{d}{dx}\left(\frac{\partial F}{\partial y^{\prime }}\right)=0$$
这是变分法的核心方程，它给出了目标函数极值所需要的条件。具体推导请看文末。

变分法的应用举例

1. 最短路径问题：求解费马原理

最经典的变分法应用之一是最短路径问题。假设我们想要计算从点 ( $A$ ) 到点 ( $B$ ) 的最短路径。在光学中，光线从一个点到另一个点总是沿着时间最短的路径传播，这就是著名的费马原理（Fermat’s Principle）。

假设光线的路径可以表示为 ( $y(x)$ )，我们希望最小化光线传播的时间 ( $T$ )。如果光线的传播速度为 ( $v$ )，则光线传播的时间是光程与速度的比值。光程的表达式为：
$$T={\int }_a^b\frac{\sqrt{1+(y^{\prime }(x){)}^2}}{v(x)}dx$$
其中，( $v(x)$ ) 表示不同点之间的光速。

现在，应用变分法，我们要求解使得 ( $T$ ) 最小化的路径 ( $y(x)$ )。这就是一个变分法问题，其拉格朗日函数为：
$$F(x,y,y^{\prime })=\frac{\sqrt{1+(y^{\prime }(x){)}^2}}{v(x)}$$
应用欧拉-拉格朗日方程，得到求解路径的必要条件。通过这个方程，我们能够求得最短路径。

2. 弹簧的振动问题

另一个常见的变分法应用是弹簧振动问题。假设我们有一个质量为 ( $m$ ) 的物体，连接在一个弹簧上，弹簧的力学特性遵循胡克定律，弹簧的恢复力与位移成正比。

系统的拉格朗日函数为：
$$L=T-U$$
其中，( $T$ ) 是动能，( $U$ ) 是势能。动能为 ( $T=\frac{1}{2}m{\dot{y}}^2$ )，势能为 ( $U=\frac{1}{2}k{y}^2$ )，其中 ( $k$ ) 是弹簧常数，( $y$ ) 是位移，( $\dot{y}$ ) 是速度。

因此，拉格朗日函数为：
$$L=\frac{1}{2}m{\dot{y}}^2-\frac{1}{2}k{y}^2$$
应用欧拉-拉格朗日方程，我们可以得到物体的运动方程：
$$m\ddot{y}+ky=0$$
这是一个标准的简谐振动方程，我们可以通过求解这个方程来得到物体的运动规律。

变分法的应用价值

变分法在许多领域都有重要应用，尤其是在物理学、工程学、经济学和计算机科学中。通过变分法，我们可以有效地求解一些复杂的优化问题，找出最优解。具体来说，变分法的应用价值主要体现在以下几个方面：

1. 物理学：在经典力学中，变分法可以用来推导出物体的运动方程，例如拉格朗日方程和哈密顿方程，这些方程是描述物理系统演化的基本方程。

2. 工程学：变分法被广泛应用于结构优化、电路设计、机械工程等领域，通过优化设计参数来实现最优的系统性能。

3. 经济学：在经济学中，变分法用于求解最优决策问题，例如最优投资策略、最优消费决策等。

4. 计算机科学：在图像处理、机器学习等领域，变分法被用来求解最优化问题，例如图像去噪、图像分割、最小化损失函数等。

总结

变分法（Calculus of Variations）是一种强大的数学工具，用于求解极值问题，尤其是在目标函数是积分形式的情况下。通过求解欧拉-拉格朗日方程，我们能够找到最优解，并应用于物理、工程、经济学等多个领域。变分法的核心思想是通过对目标函数进行变分，从而得到极值的必要条件，它在最短路径问题、弹簧振动等经典问题中都有广泛的应用。

英文版

What is the Calculus of Variations?

Calculus of Variations is a mathematical method used to find the function that makes a certain quantity attain an extremum (maximum or minimum) under specific conditions. It is primarily used in optimization problems and has applications in fields such as physics, engineering, economics, and more. The method enables us to solve seemingly complex optimization problems and find the optimal solution.

In simple terms, the goal of the calculus of variations is to find a function that, under certain constraints, minimizes or maximizes a specific quantity (usually in an integral form).

Basic Idea of the Calculus of Variations

The fundamental idea of the calculus of variations is that we are given a functional ( $J[y]$ ), which is the integral of a function over some interval ( $[a,b]$ ), and we want to find a function ( $y(x)$ ) that makes this functional attain an extremum.

We define the functional ( $J[y]$ ) as:
$$J[y]={\int }_a^{b}F(x,y(x),y^{\prime }(x))dx$$
where ( $F(x,y(x),y^{\prime }(x))$ ) is a function that depends on ( $x$ ), ( $y(x)$ ), and ( $y^{\prime }(x)$ ), known as the Lagrangian. Our goal is to use the calculus of variations to find the function ( $y(x)$ ) that minimizes or maximizes this functional.

Euler-Lagrange Equation

To solve extremum problems using the calculus of variations, we need to perform a variation on the functional and obtain the Euler-Lagrange equation, which is at the core of this method.

The derivation of the Euler-Lagrange Equation is based on an important concept: variation. Suppose ( $y(x)$ ) is the function we need to solve for, and we introduce a small variation ( $\eta (x)$ ), such that ( $y(x)$ ) becomes ( $y(x)+ϵ\eta (x)$ ), where ( $ϵ$ ) is a small parameter. We then examine how the functional changes under this variation.

We define the perturbed functional as:
$$J[y+ϵ\eta ]={\int }_a^{b}F(x,y(x)+ϵ\eta (x),y^{\prime }(x)+ϵ{\eta }^{\prime }(x))dx$$
To find the extremum, we take the derivative of this expression with respect to ( $ϵ$ ) and set it to zero, obtaining:
$$\frac{d}{dϵ}J[y+ϵ\eta ]{|}_{ϵ=0}=0$$
This results in the Euler-Lagrange Equation:
$$\frac{\partial F}{\partial y}-\frac{d}{dx}\left(\frac{\partial F}{\partial y^{\prime }}\right)=0$$
This equation provides the necessary condition for a function to make the functional attain an extremum, and it is the core of the calculus of variations.

Applications of the Calculus of Variations

1. The Shortest Path Problem: Fermat’s Principle

One of the most classic applications of the calculus of variations is the shortest path problem. Suppose we want to find the shortest path from point ( $A$ ) to point ( $B$ ). In optics, light always travels along the path that minimizes the travel time, which is known as Fermat’s Principle.

Let’s assume the path of light is ( y(x) ), and we want to minimize the time ( $T$ ) for light to travel. If the speed of light along the path is ( $v$ ), the travel time is the integral of the distance over the speed. The travel time is given by:
$$T={\int }_a^b\frac{\sqrt{1+(y^{\prime }(x){)}^2}}{v(x)}dx$$
where ( $v(x)$ ) represents the speed of light at different points.

Now, using the calculus of variations, we want to find the path ( $y(x)$ ) that minimizes ( $T$ ). This becomes a variational problem, with the Lagrangian given by:
$$F(x,y,y^{\prime })=\frac{\sqrt{1+(y^{\prime }(x){)}^2}}{v(x)}$$
By applying the Euler-Lagrange equation, we can derive the necessary condition for the path that minimizes the travel time. This equation leads us to the optimal path.

2. Spring Vibration Problem

Another common application of the calculus of variations is the spring vibration problem. Suppose we have a mass ( m ) attached to a spring, and the spring follows Hooke’s law, meaning the restoring force is proportional to the displacement.

The system’s Lagrangian function is given by:
$$L=T-U$$
where ( T ) is the kinetic energy and ( U ) is the potential energy. The kinetic energy is ( $T=\frac{1}{2}m{\dot{y}}^2$ ), and the potential energy is ( $U=\frac{1}{2}k{y}^2$ ), where ( $k$ ) is the spring constant, ( $y$ ) is the displacement, and ( $\dot{y}$ ) is the velocity.

Thus, the Lagrangian becomes:
$$L=\frac{1}{2}m{\dot{y}}^2-\frac{1}{2}k{y}^2$$
Applying the Euler-Lagrange equation gives us the equation of motion:
$$m\ddot{y}+ky=0$$
This is the standard simple harmonic oscillator equation, and by solving this equation, we can find the motion of the mass attached to the spring.

Applications and Value of the Calculus of Variations

The calculus of variations has significant applications in various fields such as physics, engineering, economics, and computer science. By using the calculus of variations, we can effectively solve complex optimization problems and find the optimal solutions. Specifically, the value of the calculus of variations lies in the following areas:

1. Physics: In classical mechanics, the calculus of variations helps derive the equations of motion for physical systems, such as the Lagrangian and Hamiltonian equations, which describe how physical systems evolve over time.

2. Engineering: In engineering, the calculus of variations is widely used in structural optimization, circuit design, mechanical engineering, etc., to achieve the best possible system performance by optimizing design parameters.

3. Economics: In economics, the calculus of variations is used to solve optimal decision problems, such as optimal investment strategies and optimal consumption decisions.

4. Computer Science: In fields like image processing, machine learning, and artificial intelligence, the calculus of variations is used to solve optimization problems, such as image denoising, image segmentation, and minimizing loss functions in training models.

Conclusion

The Calculus of Variations is a powerful mathematical tool used to solve extremum problems, particularly when the objective function is in the form of an integral. By solving the Euler-Lagrange equation, we can find the optimal solution and apply this method in various fields such as physics, engineering, economics, and computer science. The core idea of the calculus of variations is to vary the objective function and derive the necessary conditions for the extremum, and this method has broad applications in real-world optimization problems like the shortest path problem and spring vibration problems.

要详细解释这个步骤，我们需要回顾如何使用 Leibniz 法则 来对包含积分的函数进行求导。这个步骤涉及到的过程是对含有参数 ( $ϵ$ ) 的泛函进行微分。我们从以下的泛函开始：

$$J[y+ϵ\eta ]={\int }_a^{b}F(x,y(x)+ϵ\eta (x),y^{\prime }(x)+ϵ{\eta }^{\prime }(x))dx$$

欧拉-拉格朗日方程具体推导

Euler-Lagrange Equation

步骤 1：应用 Leibniz 法则

首先，我们要对 ( $J[y+ϵ\eta ]$ ) 关于 ( $ϵ$ ) 进行求导。应用 Leibniz 法则时，重要的一点是，积分上下限 ( $a$ ) 和 ( $b$ ) 是常数，并且积分的变量是 ( $x$ )，因此我们可以直接对被积函数进行求导并将导数移到积分符号外。也就是说，我们有：

$$\frac{d}{dϵ}J[y+ϵ\eta ]=\frac{d}{dϵ}\left({\int }_a^{b}F(x,y(x)+ϵ\eta (x),y^{\prime }(x)+ϵ{\eta }^{\prime }(x))dx\right)$$

根据 Leibniz 法则，可以将导数与积分交换，得到：

$$\frac{d}{dϵ}J[y+ϵ\eta ]={\int }_a^b\frac{d}{dϵ}F(x,y(x)+ϵ\eta (x),y^{\prime }(x)+ϵ{\eta }^{\prime }(x))dx$$

步骤 2：求导被积函数

接下来我们需要对被积函数 ( $F(x,y(x)+ϵ\eta (x),y^{\prime }(x)+ϵ{\eta }^{\prime }(x))$ ) 关于 ( $ϵ$ ) 进行求导。这里的被积函数是包含 ( $ϵ$ ) 的，因此我们将对其进行链式法则求导：

$$\frac{d}{dϵ}F(x,y(x)+ϵ\eta (x),y^{\prime }(x)+ϵ{\eta }^{\prime }(x))$$

按照链式法则，我们对 ( $F$ ) 的三个变量分别求导：

1. 对 ( $y(x)+ϵ\eta (x)$ ) 求导：
   $$\frac{\partial F}{\partial (y)}\cdot \frac{d}{dϵ}(y(x)+ϵ\eta (x))=\frac{\partial F}{\partial y}\cdot \eta (x)$$
   其中 ( $\eta (x)$ ) 是 ( $ϵ$ ) 的线性函数。

2. 对 ( $y^{\prime }(x)+ϵ{\eta }^{\prime }(x)$ ) 求导：
   $$\frac{\partial F}{\partial (y^{\prime })}\cdot \frac{d}{dϵ}(y^{\prime }(x)+ϵ{\eta }^{\prime }(x))=\frac{\partial F}{\partial y^{\prime }}\cdot {\eta }^{\prime }(x)$$
   同理，( ${\eta }^{\prime }(x)$ ) 是 ( $ϵ$ ) 的导数部分。

因此，综合起来，导数的表达式为：

$$\frac{d}{dϵ}F(x,y(x)+ϵ\eta (x),y^{\prime }(x)+ϵ{\eta }^{\prime }(x))=\frac{\partial F}{\partial y}\cdot \eta (x)+\frac{\partial F}{\partial y^{\prime }}\cdot {\eta }^{\prime }(x)$$

步骤 3：代入积分中

现在我们将上述导数代入积分中，得到：

$$\frac{d}{dϵ}J[y+ϵ\eta ]={\int }_a^b\left(\frac{\partial F}{\partial y}\cdot \eta (x)+\frac{\partial F}{\partial y^{\prime }}\cdot {\eta }^{\prime }(x)\right)dx$$

步骤 4：分部积分

对于第二项 ( $\frac{\partial F}{\partial y^{\prime }}\cdot {\eta }^{\prime }(x)$ )，我们将使用 分部积分法。根据分部积分法的公式：

$${\int }_a^{b}u(x)v^{\prime }(x)dx={\left[u(x)v(x)\right]}_a^b-{\int }_a^b{u}^{\prime }(x)v(x)dx$$

我们将 ( $u(x)=\frac{\partial F}{\partial y^{\prime }}$ ) 和 ( $v^{\prime }(x)={\eta }^{\prime }(x)$ )，然后对 ( $\frac{\partial F}{\partial y^{\prime }}$ ) 求导，得到：

$${\int }_a^b\frac{\partial F}{\partial y^{\prime }}\cdot {\eta }^{\prime }(x)dx={\left[\frac{\partial F}{\partial y^{\prime }}\cdot \eta (x)\right]}_a^b-{\int }_a^b\frac{d}{dx}\left(\frac{\partial F}{\partial y^{\prime }}\right)\cdot \eta (x)dx$$

因为 ( $\eta (a)=\eta (b)=0$ )，所以边界项为零。因此我们得到了：

$${\int }_a^b\frac{\partial F}{\partial y^{\prime }}\cdot {\eta }^{\prime }(x)dx=-{\int }_a^b\frac{d}{dx}\left(\frac{\partial F}{\partial y^{\prime }}\right)\cdot \eta (x)dx$$

步骤 5：最终结果

将上面的结果代入之前的表达式，我们得到：

$$\frac{d}{dϵ}J[y+ϵ\eta ]={\int }_a^b\left(\frac{\partial F}{\partial y}-\frac{d}{dx}\left(\frac{\partial F}{\partial y^{\prime }}\right)\right)\eta (x)dx$$

这就是对泛函 ( $J[y]$ ) 的关于 ( $ϵ$ ) 的导数。为了使得 ( $J[y]$ ) 在 ( $ϵ=0$ ) 处取得极值，必须要求上述积分式为零。因此，我们得到 欧拉-拉格朗日方程：

$$\frac{\partial F}{\partial y}-\frac{d}{dx}\left(\frac{\partial F}{\partial y^{\prime }}\right)=0$$

这就是变分法的核心方程，它为目标函数的极值提供了必要条件。

后记

2024年12月28日16点46分于上海， 在GPT4o大模型辅助下完成。