Question

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

Short answer

Use substitution and the Euler-Lagrange equation to derive the equation for s(t).

Step-by-step solution

Define the Function

Identify the function to be used in the integral. The given integral is: \[ \frac{dt}{s} \sqrt{s^{2} + (\frac{ds}{dt})^{2}}. \]

Identify New Variables

Introduce a new independent variable to simplify computations. Let \( u = s \) and \( v = s' = \frac{ds}{dt} \). Therefore, we want to rewrite the integral in terms of \( u \) and \( v \).

Rewrite the Function

Rewrite the integral using the new variables: \[ \int_{t_{1}}^{t_{2}} \frac{1}{s} \sqrt{s^{2} + (\frac{ds}{dt})^{2}} dt = \int_{t_{1}}^{t_{2}} \frac{1}{u} \sqrt{u^2 + v^2} \frac{du}{v}. \]

Simplify the Integrand

Consequently, we have: \[ \int_{t_{1}}^{t_{2}} \frac{1}{u} \sqrt{u^2 + v^2} \frac{du}{v}. \] Simplify this into a simpler form: \[ L = \frac{1}{u} \sqrt{u^2 + v^2}. \]

Euler-Lagrange Equation

Use the Euler-Lagrange equation to find the extremal of the integral. The Euler-Lagrange equation is given by: \[ \frac{\partial L}{\partial s} - \frac{d}{dt} \Big( \frac{\partial L}{\partial s'} \Big) = 0. \]

Compute Partial Derivatives

Compute the partial derivatives: \[ \frac{\partial L}{\partial s} = -\frac{\sqrt{u^2 + v^2}}{u^2} \] and \[ \frac{\partial L}{\partial \frac{ds}{dt}} = \frac{u^2}{u^2 + v^2}. \]

Derivative with Respect to t

Take the derivative with respect to t of \( \frac{\partial L}{\partial s'} \): \[ \frac{d}{dt} \Big( \frac{\partial L}{\partial s'} \Big) = \frac{d}{dt} \Big( \frac{u^2}{u^2 + v^2} \Big). \]

Set Equation to Zero

Set the Euler-Lagrange equation to zero and solve for s(t): \[ -\frac{\sqrt{u^2 + v^2}}{u^2} - \frac{d}{dt} \Big( \frac{u^2}{u^2 + v^2} \Big) = 0. \]

Solve for s(t)

Solve the differential equation derived in Step 8 to find \( s(t) \). The specific steps will depend on the form of the solution from above.

Key concepts

Calculus of Variations

The calculus of variations is a field of mathematical analysis used to find functions that optimize certain quantities. In this exercise, we're dealing with an integral that needs to be made stationary, meaning finding a function that either minimizes or maximizes the integral's value. To achieve this, we employ the Euler-Lagrange equation. This equation provides the necessary conditions for a function to be an extremum of the functional.
Let's break down what a functional is and how we apply it:

• A functional is essentially a function of a function, meaning it takes a function as input and produces a scalar.
• We consider small perturbations to the function and require that, up to the first order, the change in the functional is zero for an extremum to occur.

By following these steps, we transform our initial problem into a differential equation, the Euler-Lagrange equation, and solve it to find the function that satisfies our requirements.

Mathematical Physics

Mathematical physics involves applying mathematical methods to solve problems in physics. One of the key methods employed is the calculus of variations, which is fundamental in formulating physical laws.
For example, in this exercise, we are interpreting the integral as a physical quantity. Simplifying and solving it helps us understand how to describe systems in their most efficient forms.
The Euler-Lagrange equation found here is analogous to principles such as Hamilton's principle in classical mechanics, where the trajectory of a system is found by making the action integral stationary. This approach bridges the gap between abstract mathematics and practical physical problems.

• Formulations like these allow physicists to derive equations of motion.
• They also help in finding conditions under which physical systems remain stable or evolve predictably.

By mastering these equations, one gains a solid foundation for understanding more complex physical phenomena and their mathematical descriptions.

Differential Equations

Differential equations are equations involving derivatives of a function. They are fundamental in describing various physical systems, from simple motion to complex fields.
In this exercise, after identifying and rewriting the integrand, we end up with a differential equation through the Euler-Lagrange equation.
Let's understand the key steps involved in working with differential equations:

• Define your variables clearly and decide on an appropriate form for the equations.
• Introduce new variables if necessary to simplify the problem.
• Find the partial derivatives as needed, which play a crucial role in formulating the Euler-Lagrange equation.
• Solve the resulting differential equation, which often requires integration or numerical methods, to obtain the function representing your system.

For our specific case, solving the Euler-Lagrange equation involves calculating the partial derivatives of the Lagrangian and simplifying them into a solvable form.
These steps highlight the importance of differential equations in understanding and solving various mathematical and physical problems.