Extending a previous paper, we present a generalization in dimension 3 of the
traditional Szebehely-type inverse problem. In that traditional setting, the
data are curves determined as the intersection of two families of surfaces, and
the problem is to find a potential V such that the Lagrangian L = T - V, where
T is the standard Euclidean kinetic energy function, generates integral curves
which include the given family of curves. Our more general way of posing the
problem makes use of ideas of the inverse problem of the calculus of variations
and essentially consists of allowing more general kinetic energy functions,
with a metric which is still constant, but need not be the standard Euclidean
one. In developing our generalization, we review and clarify different aspects
of the existing literature on the problem and illustrate the relevance of the
newly introduced additional freedom with many examples.Comment: 23 pages, to appear in Rep. Math. Phy
