In this paper a more general definition of geodesic in \(\mathbb R^n\) with respect to a given obstacle \(U\), namely on a manifold with boundary, is considered and it is shown that a classical result, obtained by Morse and Serre for a manifold without boundary (see [\textit{J.-P. Serre}, Ann. Math. (2) 54, 425--505 (1951; Zbl 0045.26003)]) holds here, too. In fact, it is shown that, under suitable hypotheses, there exist infinite geodesics, subject to the obstacle \(U\), which join two given points. In an abstract setting the obstacle originates a nonconvex unilateral constraint on the Sobolev space \(\mathbb H^1([0,1],\mathbb R^n)\); then our geodesics are the ``stationay points'' subject to this constraint of the energy integral and we can extend, to this case, the Lusternik- Schnirelmann theory by using the notion of ``curve of maximal slope'' for a functional defined on metric spaces, given in [\textit{F. E. Browder}, Ann. Math. (2) 82, 459--477 (1965; Zbl 0136.12002); the first author and \textit{M. Tosques}, Boll. Unione Mat. Ital., VI. Ser., B 1, 143--170 (1982; Zbl 0495.58012); Existence and properties of the curves of maximal slope (to appear)].