The Brachistochrone Expanded and A Few Qeustions

This installment of “It’s Obvious. Not!” looks at:

Book: “Classical Dynamics of Particles and Systems”

Edition: third

Authors: Jerry B. Marion and Stephen T. Thornton

Publisher: Harcourt Brace Jovanovich

Pages: 176-178 (Chapter 5, Example 5.2)

Reading physics books, it often occurs to me that the authors must be aware of some patterns or 'rules of thumb' that the reader may not be privy to. Today's post expands a very truncated example from Marion and Thornton and hopefully clarifies it. This post also poses several questions in search of those patterns and rules mentioned above.

After explaining the calculus of variations and the importance of Euler's equation
Marion and Thornton follow up with a concrete example: the brachistochrone. The problem of the brachistochrone is to determine the path for a particle to move from point A to B under the influence of a constant force, (gravity for example), in the least amount of time. The 'least amount of time' phrase should key us that this is a good candidate for using Euler's Equation. In short order, the book determines that the path from point A to B can be described by the following integral equation:

the functional from this integral that should be used in Euler's equation is:

as a reminder, Euler's equation is:

The partial derivative of f with respect to y is 0. That means that the partial derivative of f with respect to y prime has to be constant. The authors decide to define that constant as one over the square root of two times the arbitrary quantity a.

By carrying out the partial differentiation from Euler's equation we arrive at, (using the chain rule of differentiation):



The authors then decide to square the result to arrive at:

This decision brings up the first question. The authors may have made this decision because they knew ahead of time that the final result for the path is a cycloid and the form of the partial differentiation shown above will lead them to that result. However, they offer no rationale as to their decision process.

Question 1: Is there a pattern that readers of the book should recognize that would lead them to square the result of the differentiation?

Next, the book solves the above equation for y, using the infamous expression 'This may be put in the form' and then stating the solution. Here are the steps



Those steps get us most of the way to the book's result. But, the last step is the tricky one and leads to the second question. The authors multiply the inside of the radical by one.


Question Number 2: Is there something about the first result for y prime that should queue the reader to multiply the inside of the radical by x over x? Is it as simple as being familiar with integration tables and realizing that the multiplication will put the integrand in a familiar form?

From here, the book solves the integral by using a change of variable:

This change of variable brings up the third question:

Question Number 3: Why did the authors chose this change of variables? It makes the solution come out as a very clean set of formulas parameterized on theta. Was there a pattern to the integral that experience reveals? Otherwise, the integral could have been worked out using integral tables and the answer would have been a rather messy expression involving an arcsin and more radicals.

Although the book performs the integral in one line, (they simply write down the answer), it's rather detailed. I'll work through the steps here.

These steps lead to one final question. If you followed the steps above, you may have come to the conclusion that:

because these two terms cancel each other out in the initial quotient.

Question Number 4: Should the equality in the above expression have been obvious? Is there a simple way to look at this, (perhaps involving a trig identity), and immediately cancel the two terms in the quotient above?

I hope I've helped clear up how the book's solution leads from step to step. I believe the key to a solid understanding of physics lies partially in the ability to recognize what direction to move the solution in, in other words, in the thought processes that aren't revealed by the book.

Do you have any thoughts on how these steps could have been simplified? Do you know why the authors made the decisions they did? Do you know an even better way to solve this? Do you other questions about the solution? If so, please comment below!