The closing decades of the 20th century have seen many scientists recognize that their mathematical models are in fact instances of DAEs, or of ODEs with invariants. Such a recognition has often carried with it the benefit of affording a new, sometimes revealing, computational look at the old problem.

But one must not conclude that reformulating a mathematical model as a DAE is always a good idea. Neither should one automatically assume that a numerical approximation of an ODE with an invariant must respect the invariant first, and only then worry about discretizing the ODE. In this talk we give some examples to the contrary:

• enforcing energy conservation in highly oscillatory Hamiltonian systems;
• moving meshes for PDEs; and
• reproducing an entire homotopy path as a way for reaching its end.

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