Question

(a) Consider the case of two dependent variables. 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 \(2[Y=y+\epsilon \eta(x) \text { with } \eta(x) \text { arbitrary }]\) and construct a similar formula for \(z\) llet \(Z=z+\epsilon \zeta(x),\) 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). (b) Consider the case of two independent variables. You want to find the function \(u(x, y)\) which makes stationary the double integral $$\int_{y_{1}}^{y_{2}} \int_{x_{1}}^{x_{2}} F\left(u, x, y, u_{x}, u_{y}\right) d x d y$$. Hint: Let the varied \(U(x, y)=u(x, y)+\epsilon \eta(x, y)\) where \(\eta(x, y)=0\) at \(x=x_{1}\) \(x=x_{2}, y=y_{1}, y=y_{2},\) but is otherwise arbitrary. As in Section \(2,\) differentiate \(x=y=y=y\), with respect to \(\epsilon,\) set \(\epsilon=0,\) integrate by parts, and use the fact that \(\eta\) is arbitrary. Show that the Euler equation is then $$\frac{\partial}{\partial x} \frac{\partial F}{\partial u_{x}}+\frac{\partial}{\partial y} \frac{\partial F}{\partial u_{y}}-\frac{\partial F}{\partial u}=0$$ (c) Consider the case in which \(F\) depends on \(x, y, y^{\prime},\) and \(y^{\prime \prime} .\) Assuming zero values of the variation \(\eta(x)\) and its derivative at the endpoints \(x_{1}\) and \(x_{2},\) show that then the Euler equation becomes $$\frac{d^{2}}{d x^{2}} \frac{\partial F}{\partial y^{\prime \prime}}-\frac{d}{d x} \frac{\partial F}{\partial y^{\prime}}+\frac{\partial F}{\partial y}=0$$

Short answer

For dependent variables y and z: \[ y: \frac{\partial F}{\partial y} - \frac{d}{dx} \left( \frac{\partial F}{\partial y'} \right) = 0 ] \and\ \[ z: \frac{\partial F}{\partial z} - \frac{d}{dx} \right) = 0 \right) = 0 \]For independent variables: \[ \frac{\partial}{\partial x} \right) \] - higher-order: \loadorder: \frac{\partial F}{\pytest_blocks}

Step-by-step solution

Define varying paths for dependent variables

For the functions y and z, construct varying paths as follows:\[ Y = y + \epsilon \eta(x) \text{ and } Z = z + \epsilon \zeta(x) \] where \eta(x) and \zeta(x) are arbitrary functions.

Substitute varying paths into functional

Substitute the varying paths into the given integral: \[ I(\epsilon) = \int_{x_1}^{x_2} F(x, Y, Z, Y', Z') \, dx \]

Compute derivative w.r.t. \epsilon

Differentiate the functional with respect to \epsilon:\[ \frac{\partial I(\epsilon)}{\partial \epsilon} = \int_{x_1}^{x_2} \left( \frac{\partial F}{\partial Y} \eta(x) + \frac{\partial F}{\partial Z} \zeta(x) + \frac{\partial F}{\partial Y'} \eta'(x) + \frac{\partial F}{\partial Z'} \zeta'(x) \right) dx \]

Integrate by parts

Integrate by parts the terms involving \eta'(x) and \zeta'(x):\[ \int_{x_1}^{x_2} \frac{\partial F}{\partial Y'} \eta' \, dx = \left[ \frac{\partial F}{\partial Y'} \eta \right]_{x_1}^{x_2} - \int_{x_1}^{x_2} \frac{d}{dx} \left( \frac{\partial F}{\partial Y'} \right) \eta \, dx \]\[ \int_{x_1}^{x_2} \frac{\partial F}{\partial Z'} \zeta' \, dx = \left[ \frac{\partial F}{\partial Z'} \zeta \right]_{x_1}^{x_2} - \int_{x_1}^{x_2} \frac{d}{dx} \left( \frac{\partial F}{\partial Z'} \right) \zeta \, dx \]

Set \epsilon=0 and simplify

Set \epsilon=0 and use the fact that the boundary terms vanish because \eta(x) and \zeta(x) are arbitrary at the endpoints. This results in:\[ \int_{x_1}^{x_2} \left( \frac{\partial F}{\partial y} - \frac{d}{dx} \left( \frac{\partial F}{\partial y'} \right) \right) \eta(x) \, dx = 0 \]\[ \int_{x_1}^{x_2} \left( \frac{\partial F}{\partial z} - \frac{d}{dx} \left( \frac{\partial F}{\partial z'} \right) \right) \zeta(x) \, dx = 0 \]

Utilize arbitrariness of \eta(x) and \zeta(x)

Since \eta(x) and \zeta(x) are arbitrary, it implies:\[ \frac{\partial F}{\partial y} - \frac{d}{dx} \left( \frac{\partial F}{\partial y'} \right) = 0 \]\[ \frac{\partial F}{\partial z} - \frac{d}{dx} \left( \frac{\partial F}{\partial z'} \right) = 0 \]These are the Euler equations for y and z.

Consider the case of two independent variables

For the functions involving two independent variables, substitute the varied path into the double integral: \[ I(\epsilon) = \int_{y_1}^{y_2} \int_{x_1}^{x_2} F(U, x, y, U_x', U_y') \, dx \, dy \]

Compute derivative w.r.t. \epsilon

Differentiate the integral with respect to \epsilon and set \epsilon=0, resulting in a similar process to Step 3, but for partial derivatives:\[ \frac{\partial I(\epsilon)}{\partial \epsilon} = \int_{y_1}^{y_2} \int_{x_1}^{x_2} \left( \frac{\partial F}{\partial U} \eta + \frac{\partial F}{\partial U_x} \eta_x + \frac{\partial F}{\partial U_y} \eta_y \right) dx \, dy \]

Integrate by parts

Integrate by parts for the terms involving \eta_x and \eta_y, and use the zero boundary conditions for \eta(x, y), which lead to: \[ \int_{x_1}^{x_2} \left( \frac{d}{dx} \left( \frac{\partial F}{\partial U_x} \right) \eta + \frac{d}{dy} \left( \frac{\partial F}{\partial U_y} \right) \eta \right) dx = 0 \]

Utilize arbitrariness of \eta(x,y)

The arbitrariness of \eta(x,y) leads to the Euler equation for two independent variables: \[ \frac{\partial}{\partial x} \left( \frac{\partial F}{\partial u_x} \right) + \frac{\partial}{\partial y} \left( \frac{\partial F}{\partial u_y} \right) - \frac{\partial F}{\partial u} = 0 \]

Consider higher-order derivatives

For Euler's equation involving higher-order derivatives: \[\begin{equation} \frac{d^2}{dx^2} \left( \frac{\partial F}{\partial y''} \right) - \frac{d}{dx} \left( \frac{\partial F}{\partial y'} \right) + \frac{\partial F}{\partial y} = 0 \]

Key concepts

calculus of variations

The calculus of variations is a field of mathematical analysis that deals with optimizing functionals. These functionals are mappings from a set of functions to the real numbers. To find extrema of these functionals, we often use techniques from differential calculus applied to function spaces. For example, in the given exercise, the goal is to make the integral \( I = \int_{x_1}^{x_2} F dx \) stationary, which means finding functions y(x) and z(x) such that small perturbations neither increase nor decrease the value of the integral.

dependent variables

Dependent variables are those that depend on the independent variable(s). In this exercise, y and z are dependent variables that depend on x. The goal is to find specific forms of these functions that make the given functional stationary. The approach involves constructing varied paths for these dependent variables: \( Y = y + \epsilon \eta(x) \) and \( Z = z + \epsilon \zeta(x) \). Here, \( \eta(x) \) and \( \zeta(x) \) are arbitrary functions that represent small perturbations in y and z.

independent variables

Independent variables are the inputs to a function or a functional that do not depend on other variables in the context of the problem. In part (a) of the exercise, x is the independent variable. However, in part (b), both x and y act as independent variables. For part (b), we define a function u(x, y) and seek to make the double integral \( \int_{y_{1}}^{y_{2}} \int_{x_{1}}^{x_{2}} F(u, x, y, u_{x}, u_{y}) dx dy \) stationary. The process for handling multiple independent variables extends techniques used in single-variable calculus of variations to multidimensional domains.

integration by parts

Integration by parts is an essential technique used in calculus, particularly in the calculus of variations, to transform integrals into forms that are easier to evaluate. The basic formula is: \[ \int u dv = uv - \int v du \]. In the context of the given exercise, integration by parts is used to handle terms involving derivatives of the variations. Specifically, terms like \( \int_{x_1}^{x_2} \frac{\partial F}{\partial y'} \eta'(x) dx \) need to be integrated by parts to remove the derivatives on \( \eta \) and \( \zeta \), resulting in boundary terms and easier-to-handle integrals.