Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 10: Problem 15

Use a Lagrange multiplier to find the curve \(x(y)\) of length \(L=\pi\) on the interval \([0,1]\) which maximizes the integral \(I=\int_{0}^{1} y(x) d x\) and pass through the points \((0,0)\) and \((1,0)\).

Short Answer

Expert verified
We formulate a function combining both the integral to be maximized and the constraint, then apply the Euler-Lagrange equation to extremize the function. Solving the differential equation and applying boundary conditions will yield the optimal curve.

Step by step solution

01

Problem Formulation

Firstly, let's write down the functional to be optimized which is \(I=\int_{0}^{1} y(x) dx\). The constraint is a fixed curve length, \(L=\pi\), which can be expressed as \(L=\int_{0}^{1} \sqrt{1+y'(x)^{2}} dx = \pi\). Using Lagrange multipliers, we combine those two equations into a function \(F\), which is \(F= y+\lambda(\sqrt{1+y'^2}-1)\), where \(\lambda\) is a yet undetermined Lagrange multiplier.
02

Deriving the Euler-Lagrange Equation

We are going to use the Euler-Lagrange equation to extremize the above functional. For a functional \(F\), the equation is \(\frac{\partial F}{\partial y} - \frac{d }{dx}(\frac{\partial F}{\partial y'}) = 0\). Apply this to \(F\) gives the equation \(1 -\frac{d }{dx} \frac{\lambda y'}{\sqrt{1+y'^2}} = 0\). After doing the derivatives and simplifying, the resulting differential equation is \(\frac{d }{dx} \frac{\lambda y'}{\sqrt{1+y'^2}} = 1\).
03

Solving the Differential Equation

Integrating each side with respect to \(x\), we obtain \(\lambda \sqrt{1+y'^2} = x + A\) where \(A\) is an integration constant. This is a first order nonlinear ordinary differential equation for the derivative \(y'(x)\). Solving this for \(y'(x)\) with the boundary condition \(y(0)=0\) will yield the final solution.
04

Determining the Lagrange Multiplier and Integrating Constant

The Lagrange multiplier \(\lambda\) and the integration constant \(A\) can be determined by using the second boundary condition \(y(1)=0\) and the constraint \(L=\pi\).
05

Final Step – Evaluating The Integral

Substituting the results of \(\lambda\) and \(A\) into the differential equation and then integrating it will yield the curve \(x(y)\) that maximizes the integral \(I\) under the given condition and through solving this, a conclusion to the exercise is found.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Find the absolute maxima and minima of the function \(f(x, y)=x^{2}+\) \(x y+y^{2}\) on the unit circle.

For each problem, locate the critical points and classify each one using the second derivative test. a. \(f(x, y)=(x+y)^{2}\). b. \(f(x, y)=x^{2} y+x y^{2}\). c. \(f(x, y)=x^{4} y+x y^{4}-x y .\) d. \(f(x, y)=x^{2}-3 x y+2 x+10 y+6 y^{2}+12\). e. \(f(x, y)=\left(x^{2}-y^{2}\right) e^{-y}\)

A particle moves under the force field \(F=-\nabla V\), where the potential function is given by \(V(x, y)=x^{3}+y^{3}-3 x y+5\). Find the equilibrium points of \(\mathbf{F}\) and determine if the equilibria are stable or unstable.

Given the cylinder defined by \(x^{2}+y^{2}=4\), find the path of shortest length connecting the given points. a. \((2,0,0)\) and \((0,2,5)\) b. \((2,0,0)\) and \((2,0,5)\)

A light ray travels from point A in a medium with index of refraction \(n_{1}\) toward point \(\mathrm{B}\) in a medium with index of refraction \(n_{2}\). Assume that the a. Write the time functional in terms of the travel path \(y(x)\). b. Apply Fermat's Principle of least time to write the Euler Equation for this functional. c. Solve the equation in part b. and show that \(n \sin \theta\) is a constant. d. Let the point at which the light is incident to the interface be at \((x, 0)\). Write an expression for the total time to travel from point \(A\) at \(\left(x_{1}, y_{1}\right)\) to point \(B\) at \(\left(x_{2}, y_{2}\right)\) in terms of the indices of refraction. and the coordinates \(x, x_{1}, x_{2}\), and \(y_{2}\). e. Treating the time as a function of \(x\), minimize this function as a function of one variable and derive Snell's law of refraction.
See all solutions

Recommended explanations on Combined Science Textbooks

Synergy

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free