On the relative category in the brake orbits problem

In this paper %dedicated to the memory of Edward Fadell and Sufian Husseini we show how the notion of the Lusternik-Schnirelmann relative category can be used to study a multiplicity problem for brake orbits in a potential well which is homeomorphic to the $N$-dimensional unit disk. The estimate of the relative category of the set of chords with endpoints on the $(N-1)$-unit sphere was shown to the third author by Fadell and Husseini while he was visiting the University of Wisconsin at Madison.