Verify if the following metric and norm satisfy the respective conditions set forth for them.
Take the first variation of the following functionals.
When I was at the Jawaharlal Nehru Planetarium last week, I happened to see an outdoor exhibit to demonstrate that the cycloidal shape takes less time than a straightline for descent due to gravity. See the photograph below. The two pipes had annular disks that slide on them and there is a provision for releasing both from the same height at the same time. When I tried this, a kid (not the one seen in the photograph) there convinced me that the straight line path is faster! In fact, it was hard to tell. Perhaps the cycloidal pipe is not perfect or may be it got bent in usage.Then, I thought may be the friction is the culprit. So,...
Pose and solve the brachistochrone problem if you take friction into account. Assume a constant coefficient of friction of 0.3 all along the path. Analytical solution may be difficult. So, solve it numerically. Juxtapose the friction-free optimal path with the solution path you get with friction. You may use the simplest model for dynamics or you may be more realistic. It is up to you.