Stephan Mertens and Sebastian Mingramm 2008 Eur. J. Phys. 29 1191 doi:10.1088/0143-0807/29/6/008
Stephan Mertens1,2 and Sebastian Mingramm1
Show affiliationsThe classical problem of the brachistochrone asks for the curve down which a body sliding from rest and accelerated by gravity will slip (without friction) from one point to another in least time. In undergraduate courses on classical mechanics, the solution of this problem is the primary example of the power of variational calculus. Here, we address the generalized brachistochrone problem that asks for the fastest sliding curve between a point and a given curve or between two given curves. The generalized problem can be solved by considering variations with varying end points. We will contrast the formal solution with a much simpler solution based on symmetry and kinematic reasoning. Our exposition should encourage teachers to include variational problems with free boundary conditions in their courses and students to try simple, intuitive solutions first.
Issue 6 (November 2008)
Received 8 July 2008, in final form 6 August 2008
Published 5 September 2008
Stephan Mertens and Sebastian Mingramm 2008 Eur. J. Phys. 29 1191
Heiko Bauke et al J. Stat. Mech. (2004) P04003
M B Stone et al 2007 New J. Phys. 9 31
D. Helbing and T. Płatkowski 2002 Europhys. Lett. 60 227
Abhijit J Chaudhari et al 2008 Phys. Med. Biol. 53 5011
Vijay Namasivayam et al 2004 J. Micromech. Microeng. 14 81
Jens Scheidtmann et al 2005 Meas. Sci. Technol. 16 119
D P M Zaks et al 2009 Environ. Res. Lett. 4 044010
M S M Saifullah et al 2002 Nanotechnology 13 659
Süleyman Ulusoy 2007 Nonlinearity 20 685