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I've been reading papers that deal with Lagrangians containing second- and higher- order derivatives of field variables. In this paper in Section 3.1, I found this very interesting quote:

The Ostrogradskian instability is instead a problem with the kinetic energy, and it manifests by the dynamical variable developing a special time dependence.

Searching for an explanation of what the special time dependence is, I've so far had no luck.

Can someone tell me what that "special time dependence" looks like?

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By the phrase "special time dependence" the Ref. 1 is effectively referring to regions $R$ in the space $S$ of solution (of the higher-order EL equation) where the corresponding (conserved) energies are "very negative" and unbounded from below, cf. the Ostrogradsky instability.

The explicit time dependence depends on the higher-derivative model. Ref. 1 offers an example in Section 3.1.

References:

  1. R. P. Woodard, The Theorem of Ostrogradsky, arXiv:1506.02210; Section 3.1.
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  • $\begingroup$ 1. That doesn't seem "special". 2. I thought he might mean oscillatory or some such. Unless I missed it, he didn't show time dependence explicitly. $\endgroup$
    – S. McGrew
    Commented Mar 10, 2019 at 18:55
  • $\begingroup$ 1. I would agree, but that's a semantic issue. 2. I updated the answer. $\endgroup$
    – Qmechanic ♦
    Commented Mar 10, 2019 at 19:39

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