Ostrogradsky instability
In applied mathematics, the Ostrogradsky instability is a feature of some solutions of theories having equations of motion with more than two time derivatives (higher-derivative theories). It is suggested by a theorem of Mikhail Ostrogradsky in classical mechanics according to which a non-degenerate Lagrangian dependent on time derivatives higher than the first corresponds to a Hamiltonian unbounded from below. As usual, the Hamiltonian is associated with the Lagrangian via a Legendre transform. The Ostrogradsky instability has been proposed as an explanation as to why no differential equations of higher order than two appear to describe physical  phenomena. However, Ostrogradsky's theorem does not imply that all solutions of higher-derivative theories are unstable as many counterexamples are known. Outline of proof. The main points of the proof can be made clearer by considering a one-dimensional system with a Lagrangian formula_1. The Euler–Lagrange equation is Non-degeneracy of formula_3 means that the canonical coordinates can be expressed in terms of the derivatives of formula_4 and vice versa. Thus, formula_5 is a function of formula_6 (if it was not, the Jacobian formula_7 would vanish, which would mean that formula_3 is degenerate), meaning that we can write  formula_9 or, inverting, formula_10. Since the evolution of formula_11 depends upon four initial parameters, this means that there are four canonical coordinates. We can write those as and by using the definition of the conjugate momentum, The above results can be obtained as follows. First, we rewrite the Lagrangian into "ordinary" form by introducing a Lagrangian multiplier as a new dynamic variable formula_16 from which, the Euler-Lagrangian equations for formula_18 read Now, the canonical momentum formula_22 with respect to formula_23 are readily shown to be while These are precisely the definitions given above by Ostrogradski. One may proceed further  to evaluate the Hamiltonian where one makes use of the above Euler-Lagrangian equations for the second equality. We note that due to non-degeneracy, we can write formula_28 as formula_29. Here, only "three" arguments are needed since the Lagrangian itself only has three free parameters. Therefore, the last expression only depends on formula_30, it effectively serves as the Hamiltonian of the "original" theory, namely, We now notice that the Hamiltonian is linear in formula_32. This is a source of Ostrogradsky instability, and it stems from the fact that the Lagrangian depends on fewer coordinates than there are canonical coordinates  (which correspond to the initial parameters needed to specify the problem). The extension to higher dimensional systems is analogous, and the extension to higher derivatives simply means that the phase space is of even higher dimension than the configuration space.