Ostrogradsky instability

In applied mathematics, the Ostrogradsky instability is a consequence of 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 linearly unstable Hamiltonian 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.

Outline of proof

The main points of the proof can be made clearer by considering a one-dimensional system with a Lagrangian $L(q,{\dot {q}},{\ddot {q}})$. The Euler-Lagrange equation is

${\frac {dL}{dq}}-{\frac {d}{dt}}{\frac {dL}{d{\dot {q}}}}+{\frac {d^{2}}{dt^{2}}}{\frac {dL}{d{\ddot {q}}}}=0.$

Non-degeneracy of $L$ means that the canonical coordinates can be expressed in terms of the derivatives of ${q}$ and vice versa. Thus, $dL/d{\ddot {q}}$ is a function of ${\ddot {q}}$ (if it was not, the Jacobian $\det[d^{2}L/(d{{\ddot {q}}_{i}}\,d{\ddot {q}}_{j})]$ would vanish, which would mean that $L$ is degenerate), meaning that we can write $q^{(4)}=F(q,{\dot {q}},{\ddot {q}},q^{(3)})$ or, inverting, $q=G(t,q_{0},{\dot {q}}_{0},{\ddot {q}}_{0},q_{0}^{(3)})$. Since the evolution of $q$ depends upon four initial parameters, this means that there are four canonical coordinates. We can write those as

$Q_{1}:=q$

$Q_{2}:={\dot {q}}$

and by using the definition of the conjugate momentum,

$P_{1}:={\frac {dL}{d{\dot {q}}}}-{\frac {d}{dt}}{\frac {dL}{d{\ddot {q}}}}$

$P_{2}:={\frac {dL}{d{\ddot {q}}}}$

Due to non-degeneracy, we can write ${\ddot {q}}$ as ${\ddot {q}}=a(Q_{1},Q_{2},P_{2})$. Note that only three arguments are needed since the Lagrangian itself only has three free parameters. By Legendre transforming, we find the Hamiltonian to be

$H=P_{1}Q_{2}-P_{2}a(Q_{1},Q_{2},P_{2})-L$

We now notice that the Hamiltonian is linear in $P_{1}$. This is Ostrogradsky's 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, which exacerbates the instability (since the Hamiltonian is linear in even more canonical coordinates).