The Regularized Free Fall I. Index Computations

The main results are, firstly, a generalization of the Conley–Zehnder index from ODEs to the delay equation at hand and, secondly, the equality of the Morse index and the clockwise normalized Conley–Zehnder index \( \mu^{\rm CZ} \). We consider the nonlocal Lagrangian action functional \( {\mathcal{B}} \) discovered by Barutello, Ortega, and Verzini [7] with which they obtained a new regularization of the Kepler problem. Critical points of this functional are regularized periodic solutions \(x\) of the Kepler problem. In this article, we look at period 1 only and at dimension one (gravitational free fall). Via a nonlocal Legendre transform regularized periodic Kepler orbits \(x\) can be interpreted as periodic solutions \((x,y)\) of a Hamiltonian delay equation. In particular, regularized \(1\)-periodic solutions of the free fall are represented variationally in two ways: as critical points \(x\) of a nonlocal Lagrangian action functional and as critical points \((x,y)\) of a nonlocal Hamiltonian action functional. As critical points of the Lagrangian action, the \(1\)-periodic solutions have a finite Morse index which we compute first. As critical points of the Hamiltonian action \( {\mathcal{A}} _ {\mathcal{H}} \), one encounters the obstacle, due to nonlocality, that the \(1\)-periodic solutions are not generated any more by a flow on the phase space manifold. Hence, the usual definition of the Conley–Zehnder index as the intersection number with a Maslov cycle is not available. In the local case, Hofer, Wysocki, and Zehnder [10] gave an alternative definition of the Conley–Zehnder index by assigning a winding number to each eigenvalue of the Hessian of \( {\mathcal{A}} _ {\mathcal{H}} \) at critical points. In this article, we show how to generalize the Conley–Zehnder index to the nonlocal case at hand. On one side, we discover how properties from the local case generalize to this delay equation, and on the other side, we see a new phenomenon arising. In contrast to the local case, the winding number is no longer monotone as a function of the eigenvalues.