Question

Show that if \(F=F\left(x, y, z, y^{\prime}, z^{\prime}\right)\), and we want to find \(y(x)\) and \(z(x)\) to make \(I=\int_{x_{1}}^{x_{2}} F d x\) stationary, then \(y\) and \(z\) should each satisfy an Euler equation as in (5.1). Hint: Construct a formula for a varied path \(Y\) for \(y\) as in Section \(2[Y=y+\epsilon \eta(x)\) with \(\eta(x)\) arbitrary \(]\) and construct a similar formula for \(z\) [let \(Z=z+\epsilon \zeta(x)\), where \(\zeta(x)\) is another arbitrary function]. Carry through the details of differentiating with respect to \(\epsilon\), putting \(\epsilon=0\), and integrating by parts as in Section 2 ; then use the fact that both \(\eta(x)\) and \(\zeta(x)\) are arbitrary to get \((5.1)\)

Short answer

To make \( I \) stationary, \( y \) and \( z \) must satisfy the Euler equations: \(\frac{\partial F}{\partial y} - \frac{d}{dx}\frac{\partial F}{\partial y'} = 0 \) and \(\frac{\partial F}{\partial z} - \frac{d}{dx}\frac{\partial F}{\partial z'} = 0 \).

Step-by-step solution

Define the varied paths

To find the stationary paths for the functions, introduce small variations. For the function y(x), let the varied path be represented as \( Y = y + \epsilon \eta(x) \), where \( \eta(x) \) is an arbitrary function and \( \epsilon \) is a small parameter. Similarly, for the function z(x), let the varied path be represented as \( Z = z + \epsilon \zeta(x) \), where \( \zeta(x) \) is another arbitrary function.

Substitute the varied paths into the integral

Substitute the varied paths into the integral \( I = \int_{x_1}^{x_2} F dx \). So, we have: \[ I(\epsilon) = \int_{x_1}^{x_2} F(y + \epsilon \eta, z + \epsilon \zeta, y', z') dx \]

Differentiate with respect to \( \epsilon \)

Differentiate \( I(\epsilon) \) with respect to \( \epsilon \) and evaluate the derivative at \( \epsilon = 0 \). Applying the chain rule, we get: \[ \frac{dI(\epsilon)}{d\epsilon} \bigg|_{\epsilon=0} = \int_{x_1}^{x_2} \left(\frac{\partial F}{\partial y} \eta + \frac{\partial F}{\partial y'} \eta' + \frac{\partial F}{\partial z} \zeta + \frac{\partial F}{\partial z'} \zeta' \right) dx \]

Integrate by parts

To eliminate the terms involving \( \eta' \) and \( \zeta' \), integrate the corresponding terms by parts: \[ \int_{x_1}^{x_2} \frac{\partial F}{\partial y'} \eta' dx = \left[\frac{\partial F}{\partial y'} \eta \right]_{x_1}^{x_2} - \int_{x_{1}}^{x_{2}} \left( \frac{d}{dx}\frac{\partial F}{\partial y'} \right) \eta dx \] Note that the boundary terms vanish because \( \eta(x_1) = \eta(x_2) = 0 \) by definition of \( \eta \). A similar expression applies to \( \zeta \).

Simplify and use the arbitrariness

Combine the results and use the fact that \( \eta(x) \) and \( \zeta(x) \) are arbitrary functions to assert that their coefficients must be zero for the integral to be stationary. This gives the Euler equations: \[ \frac{\partial F}{\partial y} - \frac{d}{dx}\frac{\partial F}{\partial y'} = 0, \quad \frac{\partial F}{\partial z} - \frac{d}{dx}\frac{\partial F}{\partial z'} = 0 \]

Key concepts

Euler-Lagrange Equation

The Euler-Lagrange equation is a fundamental equation in the calculus of variations. It is used to find the function that will make a given functional stationary. A functional is an expression that maps functions to real numbers. In this context, the functional is represented by the integral of a given function, called the Lagrangian, dependent on multiple variables and their derivatives. For example, if you have a function for y(x) and z(x), the integral of their combined function over a domain can represent a physical or geometric quantity to be optimized.
To solve for these stationary paths, you need to introduce small variations to y and z. This results in the varied paths, denoted as Y and Z, given by:
Y = y + \epsilon \e\eta(x)
Z = z + \epsilon \zeta(x)
Both \eta(x) and \zeta(x) are arbitrary functions, and \epsilon is a small parameter. By substituting these varied paths into the integral and differentiating with respect to \epsilon, you derive the Euler-Lagrange equations for y and z. The general form of these equations is:
\[ \frac{\partial F}{\partial y} - \frac{d}{dx}\frac{\partial F}{\partial y'} = 0, \quad \frac{\partial F}{\partial z} - \frac{d}{dx}\frac{\partial F}{\partial z'} = 0 \]These equations state that for the integral (the functional) to be stationary, the derivatives of the Lagrangian with respect to y, z, and their derivatives y', z' must satisfy these conditions.

Stationary Paths

Stationary paths are the solutions to the problem of finding the paths (functions) y(x) and z(x) that make the integral (also called a functional) stationary. A functional is made stationary if its first variation is zero.
Imagine you have a path y(x) and a slight variation \eta(x) is added to it. This results in a varied path Y = y + \epsilon\eta(x). The same process is done for z(x). By substituting these varied paths into the functional's integral and differentiating with respect to \epsilon,
\[ I(\epsilon) = \int_{x_1}^{x_2} F(y + \epsilon\eta, z + \epsilon\zeta, y', z') dx \]and evaluating the result at \epsilon=0, we ensure the functional's value doesn’t change for small variations in y and z. Integrating by parts helps eliminate higher-order derivative terms, reducing the equation to forms dependent solely on y, z, and their first derivatives.
The key is that since \eta(x) and \zeta(x) are arbitrary, the terms multiplying them must be zero independently. This leads to the Euler-Lagrange equations that determine the stationary paths.

Functionals

In the context of calculus of variations, a functional is an integral that depends on a function and possibly its derivatives. For example, consider an integral of the form:
\[ I = \int_{x_1}^{x_2} F(y, y', z, z', x) dx \]Here, F is a function involving y, y', z, z', and x. Functionals map functions to real numbers and aim at finding the functions y(x) and z(x) that make this integral stationary.
Think of functionals as a way of encoding a problem where you're looking to optimize something over a continuous domain. Usage of functionals is widespread in physics, engineering, and economics.
To solve a problem involving functionals, we introduce small variations to the functions involved, differentiate with respect to a small parameter, and set this derivative to zero. This methodology derives the Euler-Lagrange equations, providing the necessary conditions for the integral to be stationary.
This fundamental concept allows complex real-world problems to be formulated and solved within the elegant mathematical framework of calculus of variations.