Functional topology

Introduction. The present paper gives the fundamental existence proof for a homotopic critical point under hypotheses which are less stringent than those employed in the author's recent fascicule on functional topology and abstract variational theory [2], This relaxation of hypotheses seems to be necessary if one is to extend the variational theory in the large to non-regular problems. Non-regular integrals include some of the most important and interesting integrals such as the Jacobi least action integral 1 in the three body problem of celestial mechanics. The author believes that a topological basis for the planetary orbits will be disclosed by studying the contour manifolds of this Jacobi integral.