The brachistochrone is a well-known problem in calculus of variations and optimal control. I ask if there is any explicit solution to the problem?. In the existing texts, x and y are usually parameterized in dummy variables as t, phi or theta. I want to know whether any explicit solution is available or not? I mean y(x) in terms of x, not any other intermediary variable.
In the pdf, attached to this question, there is some hint on the explicit solution (but it seems it is computational and numerical).
https://en.wikipedia.org/wiki/Brachistochrone_curve
http://mathworld.wolfram.com/BrachistochroneProblem.html
Related to your query, I suggest http://mathshistory.st-andrews.ac.uk/HistTopics/Brachistochrone.html
https://www.maa.org/sites/default/files/Brookfield-MMz-201034137.pdf
Mr. doctor Muhammad Ali , thank you. Good resources with concise info. But still it seems there is no explicit solution in your pdf and as well in the linked URL. In the pdf, only an explicit solution is presented for the catenary problem as y=y(x).
For the brachistochrone problem, the solution is still parameterized in terms of theta. Anyway, thank you again. Good resources.
Dear Saeb,
I suggest you to see link and book in attached file, it seems to me that you have an explicit solution of this problem.
https://www.springer.com/gp/book/9783030061722
Best regards
Mr. professor Mohamed-Mourad Lafifi. I have had a look on the books you referred me to. In the first book, brachistochrone problem is mentioned in the introduction section only. In the second book, the solution is only in ordinary differential equation (ode) format, as p=sqrt(2cgx/(1-2cgx)), where p=y'=dy/dx.
It seems there were no analytic explicit solution as y(x) in terms of x, out there. Anyway, thank you for your resources.
It seems somewhat explicit analytic solution is presented in this book:
Introduction to the calculus of variations, by William Elwood Byerly.
Equation (2) of chapter II.
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