Question

The shape of a hanging chain between the points \((-a, b)\) and \((a, b)\) is such that the gravitational potential energy $$ V[y]=\rho g \int_{-a}^{a} y \sqrt{1+y^{\prime 2}} d x $$ is minimized subject to the length of the chain remaining constant, $$ L[y]=\int_{-a}^{a} \sqrt{1+y^{\prime 2}} d x $$ Find the shape, \(y(x)\) of the hanging chain.

Short answer

The shape of the hanging chain, \(y(x)\), is described by the function \(y(x)=b \cosh(\frac{x}{b})\).

Step-by-step solution

Formulate the Functional that Characterizes the Problem

Consider the energy functional \(V[y]\), which is a function of the potential energy associated with the hanging chain's position \(y\) and its derivative \(y'\). Also, the integral constraint refers to the fact that the length of the chain remains constant \(L[y]\). We will write the Langrange function to include this constraint: \(F(y,y',\lambda)= y \sqrt{1+y^{\prime 2}} - \lambda \sqrt{1+y^{\prime 2}}\)

Obtain the Euler-Lagrange Equation

The necessary condition for \(y(x)\) to minimize the functional is given by Euler's Equation. It is obtained by taking the partial derivative of \(F\) with respect to \(y\), the partial derivative of \(F\) with respect to \(y'\), and combining these results into a differential equation: \(\frac{\partial F}{\partial y}-\frac{d}{dx}(\frac{\partial F}{\partial y'})=0\).

Solve the Euler-Lagrange Equation

Solving the Euler-Lagrange equation will result in a differential equation that \(y(x)\) must satisfy. We can solve this differential equation to find the shape of the hanging chain: \(y(x)=b \cosh(\frac{x}{b})\).

Key concepts

Gravitational Potential Energy

In physics, gravitational potential energy represents the energy an object possesses because of its position in a gravitational field. It is the energy that could be converted into kinetic energy if the object were allowed to fall.

For a chain hanging between two points, the overall gravitational potential energy depends on the shape the chain takes. The equation provided in the exercise,
\[ V[y]=\rho g \int_{-a}^{a} y \sqrt{1+y'^{2}} dx \],
captures this concept by integrating the height, y, of each small segment of the chain along its length, taking into account its mass per unit length ρ and the acceleration due to gravity g. The term under the square root accounts for the length of each differential segment of the chain.

Euler-Lagrange Equation

The Euler-Lagrange equation is a fundamental equation in the calculus of variations, a branch of mathematical analysis dealing with the optimization of functionals. A functional is a map from a space of functions to the real numbers, typically representing some physical quantity like energy.

In the step by step solution, the Euler-Lagrange equation is used to find the state that minimizes the functional:
\[ \frac{\partial F}{\partial y} - \frac{d}{dx}(\frac{\partial F}{\partial y'})=0 \].
This equation yields the condition that must be satisfied by the function y(x) that minimizes the functional. Here, F represents the Lagrangian which is a function of y, its derivative y', and potentially an additional parameter λ for constraints such as the chain's length.

Variational Principles

Variational principles form the core of many physical theories and mathematical problems. These principles state that some physical quantities are at their extreme values when a system is in equilibrium. To be specific, in mechanics, systems tend to an arrangement that minimizes potential energy.

In the case of the hanging chain problem, we look for a curve y(x) for which the functional representing gravitational potential energy is at its minimum value. This means the actual shape of the chain is such that the total potential energy in the gravitational field is as low as possible, consistent with the constraint of the chain's fixed length.

Functional Optimization

Functional optimization involves finding the function that minimizes or maximizes a given functional. In other words, unlike in classical optimization, where we look for the minimum of some function f(x), in functional optimization, we are concerned with finding the function y(x) that minimizes a functional V[y].

In this context, the hanging chain is an example where we apply functional optimization to determine the shape that minimizes the gravitational potential energy under the constraint of a constant length. This kind of problem is brilliantly catered to by incorporating the optimizing condition into the Euler-Lagrange equation and solving for y(x), resulting in a specific mathematical expression for the chain's equilibrium shape.