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Ostrogradsky's instability theorem says that under some conditions, a system governed by a Lagrangian which depends on time derivatives beyond the first is "unstable". In the proof, one computes the Hamiltonian and shows that the dependence on one of the canonical momenta is linear.

I don't understand either the statement or the proof. I think this is because I don't remember my Hamiltonian mechanics very well.

Questions:

  1. What is meant by saying that the system is unstable? There are several notions of stability out there. Is the theorem saying the for any choice of initial conditions, the system is, say, Lyapunov unstable?

  2. What does the linearity of the Hamiltonian have to do with stability?

  3. My sense is that Ostrogradsky's theorem is sometimes taken to justify the idea that Lagrangians in physics shouldn't depend on higher time derivatives. Why is this? What's wrong with studying systems which are unstable? If I understand what "unstable" means, isn't it just another word for "chaotic"? And certainly there are chaotic systems in the real world...

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1 Answers1

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Not a specialist here either, but I think the Scholarpedia page linked in the OP already provides the answers.

  1. What is meant by saying that the system is unstable? There are several notions of stability out there. Is the theorem saying the for _any_ choice of initial conditions, the system is, say, Lyapunov unstable?

It's unstable in the sense of presenting explosive vacuum decay:

In fact the system instantly evaporates into a maelstrom of positive and negative energy particles.

  1. What does the linearity of the Hamiltonian have to do with stability?

    The root of the problem seems to be that:

Because the Hamiltonian is linear in all but one of the conjugate momenta it is possible to arbitrarily increase or decrease the energy by moving different directions in phase space.

A similar explanation is found in this answer:

$H$ has only a linear dependence on $P_1$, and so can be arbitrarily negative. In an interacting system this means that we can excite positive energy modes by transferring energy from the negative energy modes, and in doing so we would increase the entropy — there would simply be more particles, and so a need to put them somewhere. Thus such a system could never reach equilibrium, exploding instantly in an orgy of particle creation.

  1. My sense is that Ostrogradsky's theorem is sometimes taken to justify the idea that Lagrangians in physics _shouldn't_ depend on higher time derivatives. Why is this? What's wrong with studying systems which are unstable? If I understand what "unstable" means, isn't it just another word for "chaotic"? And certainly there are chaotic systems in the real world...

It's certainly not because one wants to avoid chaotic systems, but rather that

[explosive vacuum decay] certainly does not describe the universe of human experience in which all particles have positive energy and empty space remains empty.

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