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The Ostrogradsky instability theorem says that if a Lagrangian depends on more than the position and velocity, the corresponding Hamiltonian is unbounded below. This has been suggested as a reason why you don't generally see PDEs beyond order 2.

However, most references are from physics and the rigor is lacking a bit. Is there a rigorous introduction to the Ostrogradsky instability theorem from a mathematician's perspective?

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    $\begingroup$ I've added the "classical mechanics" tag. People who work on it might know the answer. $\endgroup$
    – algori
    Commented Jul 20 at 20:37
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    $\begingroup$ It might be helpful if you could tell us in what respect rigor is missing in the usual proof of the fact that the Hamiltonian corresponding to a Lagrangian $L(q,\dot{q},\ddot{q})$ is linear in $\partial L/\partial \ddot{q}$ and hence is unbounded, for example as given on page 1 of arXiv:2001.02483. Note that this not yet refer to the appearance of an instability. That goes beyond Ostrogradsky, is not part of the theorem, and is not true in general (counterexamples are in arXiv:2403.19777). $\endgroup$
    – Carlo Beenakker
    Commented Jul 20 at 20:44

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On the problem of stability for higher-order derivative Lagrangian systems in Letters in Mathematical Physics (1987) may have the desired level of rigor (see Theorem 1).

The proof of the theorem is a straightforward few lines of algebra, so I would think that most derivations in the literature have the required level of rigor. All of this refers to the basic fact that higher-than-first-order time derivatives in the Lagrangian produce linear momentum terms in the Hamiltonian. The conclusion that this implies an instability goes beyond Ostrogradsky, is not part of the theorem, and is not true in general. (See arXiv:2403.19777 for counter examples.)

A major line of research is to find conditions on the Lagrangian that allow for higher order derivatives without compromising the stability, see Ghost-free theories with arbitrary higher-order time derivatives.

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