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Chapter 9: Problem 3

Find a first integral of the Euler equation to make stationary the integrals. $$\int_{a}^{b} \frac{y y^{\prime 2} d x}{\sqrt{1+y^{\prime 2}}}$$

Short Answer

Expert verified
\[ \frac{2y y'}{(1 + (y')^2)^{3/2}} = C \]

Step by step solution

01

Formulate the Lagrangian

Identify the integrand in the given integral as the Lagrangian, which is a function of y and y'. Here, the Lagrangian is: \ \[ L(y, y') = \frac{y (y')^2}{\sqrt{1 + (y')^2}} \ \]
02

Euler-Lagrange Equation

Write down the Euler-Lagrange equation specific to the given Lagrangian: \ \[ \frac{\text{d}}{\text{d}x} \frac{\text{d}L}{\text{d}y'} - \frac{\text{d}L}{\text{d}y} = 0 \ \]
03

Compute the Partial Derivatives

Compute the partial derivatives: \ \[ \frac{\text{d}L}{\text{d}y} = \frac{(y')^2}{\sqrt{1 + (y')^2}} \ \] \ \[ \frac{\text{d}L}{\text{d}y'} = \frac{2y y' \sqrt{1 + (y')^2} - y (y')^3 \frac{1}{2 (1 + (y')^2)^{-1/2}} }{(1 + (y')^2)^{3/2}} = \frac{2y y'}{(1 + (y')^2)^{3/2}} \ \]
04

Substitute into Euler-Lagrange Equation

Substitute the partial derivatives into the Euler-Lagrange equation to get: \ \[ \frac{\text{d}}{\text{d}x} \left( \frac{2y y'}{(1 + (y')^2)^{3/2}} \right) - \frac{(y')^2}{\sqrt{1 + (y')^2}} = 0 \ \]
05

First Integral

The equation simplifies because the term involving \(\frac{\text{d}}{\text{d}x}\) suggests that the expression inside the parenthesis must be a constant: \ \[ \frac{2y y'}{(1 + (y')^2)^{3/2}} = C \ \] where C is a constant.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

calculus of variations
The calculus of variations is a field of mathematical analysis that deals with optimizing functionals. Functionals are mappings from a set of functions to the real numbers. In simpler terms, it means finding a function that makes the value of an integral the largest or smallest possible.

In our exercise, we aim to make a specific integral stationary, meaning we want to find the function that either maximizes or minimizes this integral value. This involves finding a function that satisfies the Euler-Lagrange equation, which is a necessary condition for an extremum in calculus of variations. By solving this, we can deduce the first integral that makes our integral stationary.
first integral
A first integral simplifies our original differential equation and helps us solve for the necessary conditions. In Lagrangian mechanics, a first integral can often be related to conservation laws, such as the conservation of energy or momentum.

For our problem, we reduce the complex Euler-Lagrange equation to a simpler form involving a constant. This simpler form is referred to as a first integral. Specifically, we identify: \( \frac{2y y'}{(1 + (y')^2)^{3/2}} = C \), where C is a constant. Recognizing this first integral is crucial because it gives us a direct route to solving our problem.
Lagrangian mechanics
Lagrangian mechanics is a reformulation of classical mechanics introduced by Joseph Louis Lagrange. The main idea is to derive the equations of motion using the Lagrangian, a function that summarizes the dynamics of the system.

The Lagrangian is typically defined as the kinetic energy minus the potential energy. In variational problems, we use the Lagrangian to derive the Euler-Lagrange equation, which provides the conditions necessary for finding the function that optimizes our integral.

For our specific problem, we identified the Lagrangian as: \( L(y, y') = \frac{y (y')^2}{\text{sqrt}(1 + (y')^2)} \). This function is fundamental in setting up the conditions for stationarity.
partial derivatives
Partial derivatives are essential in multivariable calculus. They measure how a function changes as one of its input variables changes, while keeping the other variables constant.

In the context of the Euler-Lagrange equation, we need to take partial derivatives of the Lagrangian with respect to both the function y and its derivative y'. This helps set up the equation used to find the function that makes our integral stationary.

For our problem, the partial derivatives were computed as: \( \frac{\text{d}L}{\text{d}y} = \frac{(y')^2}{\text{sqrt}(1 + (y')^2)} \) and \( \frac{\text{d}L}{\text{d}y'} = \frac{2y y'}{(1 + (y')^2)^{3/2}} \). Calculating these derivatives correctly is crucial for accurately solving the Euler-Lagrange equation.

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Most popular questions from this chapter

(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 $$\frac{d^{2}}{d x^{2}} \frac{\partial F}{\partial y^{\prime \prime}}-\frac{d}{d x} \frac{\partial F}{\partial y^{\prime}}+\frac{\partial F}{\partial y}=0$$

Set up Lagrange's equations in cylindrical coordinates for a particle of mass \(m\) in a potential field \(V(r, \theta, z) .\) Hint: \(v=d s / d t ;\) write \(d s\) in cylindrical coordinates.

Prove that a particle constrained to stay on a surface \(f(x, y, z)=0,\) but subject to no other forces, moves along a geodesic of the surface. Hint: The potential energy \(V\) is constant, since constraint forces are normal to the surface and so do no work on the particle. Use Hamilton's principle and show that the problem of finding a geodesic and the problem of finding the path of the particle are identical mathematics problems.

Write and solve the Euler equations to make the following integrals stationary. Change the independent variable, if needed, to make the Euler equation simpler. \(\int_{t_{1}}^{t_{2}} s^{-1} \sqrt{s^{2}+s^{\prime 2}} d t, \quad s^{\prime}=d s / d t\)

A uniform flexible chain of given length is suspended at given points \(\left(x_{1}, y_{1}\right)\) and \(\left(x_{2}, y_{2}\right) .\) Find the curve in which it hangs. Hint: It will hang so that its center of gravity is as low as possible.
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