# Help with Setting Up/Simplifying Euler-Lagrangian

• stephenklein
In summary, the conversation discusses working through parts (a) and (b) and obtaining partial derivatives for part (c). The speaker simplifies their result to match a Wolfram widget, but notes that it differs slightly from the given answer. They also mention an unspecified area that is to be maximized in a problem involving a string with movable endpoints.
stephenklein
Homework Statement
Imagine we have a string of length ##l## in the ##x y## plane with endpoints ##x=-a## and ##x=a##. The endpoints can move to maximize this area, but the total length of the string is fixed.

a) Show that ##dx = \sqrt {1-y'^2}## where ##ds## is a tiny length of the curve such that ##ds = \sqrt {dx^2+dy^2}## as in class.

b) The area of the shaded rectangle above is ##ydx##, so the sum of all of those areas between ##x=-a## and ##x=a## gives us the total area. However, it’s easier to incorporate the fact that the length is fixed as ##l## by integrating with respect to s, the path length of moving along the string instead. Using the expression in part (a) to convert the ##ydx## integral to an integral with respect to ##ds##, express the area under the string.

c) Using the Euler-Lagrange equations prove that the optimal shape of the string is a semicircle.

A couple hints:

1) The Euler-Lagrange equation should yield ##\frac {dy} {ds} = \sqrt {1- \frac {y^2} {C^2}}##, which you can solve using separation of variables (and probably looking up the integral).

2) A semicircle of radius R has the equation ##x^2+y^2=R##
Relevant Equations
Euler-Lagrange equation: $$\frac {\partial f} {\partial y} - \frac {d} {ds} \frac {\partial f} {\partial y'} = 0$$
I was able to work through parts (a) and (b). For part (c), I got $$\frac {\partial f} {\partial y} = \sqrt {1-y'^2}$$ and $$\frac {\partial f} {\partial y'} = \frac {-y y'} {\sqrt {1-y'^2}}$$ Taking ##\frac {d} {ds}## of the latter, I used the product rule for all three terms ##y, y', (\sqrt{1-y'^2})^{-1/2}## and my result was $$\sqrt {1-y'^2} + \frac {y y''+y'^2} {\sqrt {1-y'^2}} + \frac {y y'^2 y''} {(1-y'^2)^{3/2}} = 0$$ I'm unsure (even more, skeptical) that this result simplifies to the one given in the question. I'm confident in everything up until the last derivative, which has a lot of moving parts. Any thoughts?

EDIT: I simplified the above result to ##y'=\sqrt {1+y y''}##, which agrees with a Wolfram widget I found that simplifies Lagrangian equations. I feel like I'm very close, but my result has a sum instead of a difference, and that pesky ##y''## term isn't in the given answer.

Last edited:
stephenklein said:
Homework Statement: Imagine we have a string of length ##l## in the ##x y## plane with endpoints ##x=-a## and ##x=a##. The endpoints can move to maximize this area
What area? You have only told us about a string of length ##l##. Which area is to be maximised?

## What is Euler-Lagrangian?

Euler-Lagrangian is a mathematical technique used to solve problems involving the motion of physical systems. It is based on the principle of least action, which states that a system will follow a path that minimizes the total action of the system.

## Why is Euler-Lagrangian important?

Euler-Lagrangian is important because it allows us to describe and solve problems involving the motion of physical systems in a mathematical and systematic way. It has applications in various fields such as physics, engineering, and economics.

## How do I set up an Euler-Lagrangian problem?

To set up an Euler-Lagrangian problem, you need to first identify the relevant variables, such as position, velocity, and time, and determine the Lagrangian of the system. The Lagrangian is a function that describes the energy of the system. Then, you can use the Euler-Lagrange equations to solve for the equations of motion.

## Can Euler-Lagrangian be simplified?

Yes, Euler-Lagrangian can be simplified by using symmetries and conserved quantities of the system. This can reduce the number of equations and make the problem easier to solve.

## What are some common mistakes when using Euler-Lagrangian?

Some common mistakes when using Euler-Lagrangian include forgetting to include all the relevant forces in the Lagrangian, not using the correct boundary conditions, and making errors in the derivation of the Euler-Lagrange equations. It is important to double-check your work and make sure all steps are correct.

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