  The numerical simulation of realistic reactive flows is a major challenge due
to the stiffness and high dimension of the corresponding kinetic differential
equations. Manifold-based model reduction techniques address this problem by
projecting the full phase space onto manifolds of slow motion, which capture
the system's long-term behavior. In this article we study the trajectory-based
optimization approach by Lebiedz (2004), which determines these manifolds as
minimizers of an appropriate entropy functional. Similar to other methods in
this field, this approach is based on physical and geometric intuition and was
tested on several models. This article provides a rigorous explanation for its
effectiveness, showing how it approximates nonuniformly normally attracting
orbits. It also outlines how the method can be utilized to approximate
nonuniformly normally attracting invariant manifolds of higher dimension.
Throughout the article we use a coordinate-free formulation on a Riemannian
manifold. This is especially useful for systems subject to nonlinear
constraints, e.g., adiabatic constraints.
