A simple way to solve the brachistochrone problem with resistance

Archishman Raju

Department of Physics St. Stephen’s College Delhi 110007, India.

Variational problems are ubiquitous in physics. But an introductory course on the calculus of variations is typically restricted to solving a few standard problems like the classical brachistochrone. Several experiments have clearly shown this theory to be inadequate, because any actual physical situation involves resistance, but no attempt has been made so far to reconcile experiment with theory. Adding resistive forces to the problem makes analytical solutions intractable. We show how such hard variational problems can be easily solved using a simple numerical approach. This allows a large variety of variational problems to be solved at an introductory level and the solution checked against simple experiments. We illustrate this by solving the brachistochrone problem with Coulomb friction and fluid resistance. We outline an experiment which could be used to check the result.