  Chemical kinetic models in terms of ordinary differential equations
correspond to finite dimensional dissipative dynamical systems involving a
multiple time scale structure. Most dimension reduction approaches aimed at a
slow mode-description of the full system compute approximations of
low-dimensional attracting slow invariant manifolds and parameterize these
manifolds in terms of a subset of chosen chemical species, the reaction
progress variables. The invariance property suggests a slow invariant manifold
to be constructed as (a bundle of) solution trajectories of suitable ordinary
differential equation initial or boundary value problems. The focus of this
work is on a discussion of fundamental and unifying geometric and analytical
issues of various approaches to trajectory-based numerical approximation
techniques of slow invariant manifolds that are in practical use for model
reduction in chemical kinetics. Two basic concepts are pointed out reducing
various model reduction approaches to a common denominator. In particular, we
discuss our recent trajectory optimization approach in the light of these two
concepts. We relate both of them in a variational boundary value viewpoint,
propose a Hamiltonian formulation and conjecture its relation to conservation
laws, (partial) integrability and symmetry issues as underlying fundamental
principles and potentially unifying elements of diverse dimension reduction
approaches.
