Generalization of the least uncomfortable journey problem

Abstract

The variational problem of the least uncomfortable journey between two locations on a straight line is simplified by a choice of the dependent variable. It is shown that taking the position, instead of the velocity, as the optimal function of time to be determined does away with the isoperimetric constraint. The same results as those found with the velocity as the dependent variable are obtained in a simpler and more concise way. Next the problem is generalized for motion on an arbitrary curve. In the case of acceleration-induced discomfort, it is shown that, as expected, motion on a curved path is always more uncomfortable than motion on a straight line. It is not clear that this is necessarily the case for jerk-induced discomfort, which appears to indicate that the acceleration provides a more reasonable measure of the discomfort than the jerk. The example of motion on a circular path is studied. Although we have been unable to solve the problem analytically, approximate solutions have been constructed by means of trial functions and the exact solution has been found numerically for some choices of the relevant parameters.

Keywords:
Calculus of variations; higher-derivative variational problem; free endpoints and boundary conditions; generalized least uncomfortable journey problem