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 of 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.

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

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

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