Problem 7 - Calculus Of Variations, With Applications To Physics And Engineering

Question

(a) Derive the differential equation satisfied by the four-times- differentiable function \(y(x)\) which extremizes the integral $$ I=\int_{x_{1}}^{x_{1}} f\left(x, y, y^{\prime}, y^{\prime \prime}\right) d x $$ under the condition that both \(y\) and \(y^{\prime}\) are prescribed at \(x_{i}\) and \(x_{2}\). (b) Show that, if neither \(y\) nor \(y^{\prime}\) is prescribed at either end point, the conditions $$ \frac{\partial f}{\partial y^{\prime \prime}}=0, \quad \frac{\partial f}{\partial y^{\prime}}-\frac{d}{d x}\left(\frac{\partial f}{\partial y^{\prime \prime}}\right)=0 $$ must be met at \(x=x_{1}\) and \(x=x_{2}\). (c) Generalize the result of \((a)\) by supposing $$ f=f\left(x, y, y^{\prime}, \ldots, y^{(n)}\right) $$ where \(y^{(n)}\) designates the \(n\)th derivative of \(y(x)\).

Step-by-Step Solution

Answer

The obtained differential equation in part (a) is a fourth order differential equation which extremizes the integral I, and involves the derivatives of y up to y''. For part (b), the given conditions at \(x_{1}\) and \(x_{2}\) are obtained from the variational problem when the variations \(\delta y\) and \(\delta y'\) are unrestricted at the end points \(x_{1}\) and \(x_{2}\). In part (c), the generalization for an arbitrary order derivative leads to a 2n-th order differential equation which must be satisfied by y(x) in order to extremize the integral I.

Step 1: Part a: Euler-Lagrange equation

Euler-Lagrange equation will be used, which is an vital tool when dealing with calculus of variations problems. The general form of the equation is given by \(\frac{\partial f}{\partial y}- \frac{d}{dx} \frac{\partial f}{\partial y^{\prime}}+ \frac{d^2}{dx^2} \frac{\partial f}{\partial y^{\prime \prime}} =0\). In this case, f depends on \(x, y, y'\) and \(y''\), so the Euler-Lagrange equation will provide a fourth order differential equation that must be satisfied by \(y(x)\).

Step 2: Part b: Additional conditions

For this part, neither \(y\) nor \(y'\) is prescribed at the end points \(x_{1}\) and \(x_{2}\). This alters the boundary conditions of the problem so the variation \(\delta y\) and its first derivative \(\delta y'\) are now allowed to be non-zero at the end points. Taking this into consideration, the terms involving the variation \(\delta y''\) must vanish at the end points, leading to the system of equations provided in the problem. The first equation is \(\frac{\partial f}{\partial y^{\prime \prime}}=0\) at \(x_{1}\) and \(x_{2}\) and the second one is \(\frac{\partial f}{\partial y^{\prime}}-\frac{d}{d x}\left(\frac{\partial f}{\partial y^{\prime \prime}}\right)=0\) at \(x_{1}\) and \(x_{2}\).

Step 3: Part c: Generalization to n-th derivative

Generalizing the result of part (a) to an integral \(I\) which depends on the \(n\)th derivative of \(y(x)\), denoted by \(y^{(n)}\), implies the use of variational calculus for higher order derivatives. The Euler-Lagrange equations must be modified to account for this higher derivative. The result will be a 2n-th order differential equation for \(y(x)\), which must be satisfied in order to extremize the integral \(I\).

Key Concepts

Calculus of Variations

When tackling problems in the calculus of variations, we explore how to find functions that make certain integrals achieve their maximum or minimum values. Imagine wanting to find a path or shape that makes an expression like \( I = \int_{x_{1}}^{x_{2}} f(x, y, y', y'') \, dx \) either as small as possible or as big as possible. This is where the concept of extremizing an integral comes in.

• The main tool we use here is the Euler-Lagrange equation, which transforms this problem into finding a differential equation that the function must satisfy.
• We consider variations of the function, meaning small tweaks, to see how the integral is affected.
• If setting a small change in the function, \( \delta y \), leads to no increase or decrease in the value of the integral, we've likely found a path that extremizes it.

The application of calculus of variations is broad, finding uses in fields from physics to economics, where optimal solutions are sought for various systems.

Higher Order Differential Equations

The Euler-Lagrange equation can give rise to higher order differential equations, especially when our function depends on higher derivatives like \( y'' \).
For instance, in part (a) of our exercise, where \( f \) depends on \( x, y, y', \) and \( y'' \), the resultant Euler-Lagrange equation forms a fourth order differential equation:

• This means the function \( y(x) \) must be four-times-differentiable, leading us to check how it behaves upon differentiating multiple times.
• Solving these types of equations often involves advanced techniques and boundary conditions that frame how the solution behaves.

Higher order differential equations are more complex than second order ones due to increased terms and complexity, but they are vital for accurately modeling phenomena like beam deflections, fluid dynamics, and many other systems.

Boundary Conditions in Optimization

Boundary conditions play a crucial role in solving problems where we want to optimize functions. They define the values or behavior of the function at the edges, such as at \( x = x_1 \) and \( x = x_2 \), and they impact how the solution evolves.
In part (b) of our exercise, the conditions are adjusted when neither \( y \) nor \( y' \) are prescribed.

• This means that the solution must satisfy the given equations at the endpoints, making the choice of boundary conditions critical for the optimization process.
• The conditions \( \frac{\partial f}{\partial y''} = 0 \) and \( \frac{\partial f}{\partial y'} - \frac{d}{dx}\left( \frac{\partial f}{\partial y''} \right) = 0 \) inform us about the necessary relationships that must be upheld at the boundaries.

By ensuring that these conditions are met, we guarantee that the variations do not cause unintended changes in the value of the integral across the interval, helping us find the desirable function more efficiently.