Ostrogradsky instability

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

${\displaystyle {\frac {\partial L}{\partial q}}-{\frac {d}{dt}}{\frac {\partial L}{\partial {\dot {q}}}}+{\frac {d^{2}}{dt^{2}}}{\frac {\partial L}{\partial {\ddot {q}}}}=0.}$

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, ${\displaystyle \partial L/\partial {\ddot {q}}}$ is a function of ${\displaystyle {\ddot {q}}}$ (if it were not, the Jacobian ${\displaystyle \det[\partial ^{2}L/(\partial {{\ddot {q}}_{i}}\,\partial {\ddot {q}}_{j})]}$ 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

${\displaystyle Q_{1}:=q}$

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

and by using the definition of the conjugate momentum,

${\displaystyle P_{1}:={\frac {\partial L}{\partial {\dot {q}}}}-{\frac {d}{dt}}{\frac {\partial L}{\partial {\ddot {q}}}}}$

${\displaystyle P_{2}:={\frac {\partial L}{\partial {\ddot {q}}}}}$

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 ${\displaystyle \lambda }$

${\displaystyle L(q,{\dot {q}},{\ddot {q}})\to {\tilde {L}}=L(Q_{1},{\dot {Q_{1}}},{\dot {Q_{2}}})-\lambda (Q_{2}-{\dot {Q_{1}}})}$,

from which, the Euler-Lagrangian equations for ${\displaystyle Q_{1},Q_{2},\lambda }$ read

${\displaystyle Q_{1}:{\frac {d}{dt}}{\frac {\partial L}{\partial {\dot {Q_{1}}}}}+{\dot {\lambda }}-{\frac {\partial L}{\partial Q_{1}}}=0}$,

${\displaystyle Q_{2}:{\frac {d}{dt}}{\frac {\partial L}{\partial {\dot {Q_{2}}}}}+{\lambda }=0}$,

${\displaystyle \lambda :Q_{2}-{\dot {Q_{1}}}=0}$,

Now, the canonical momentum ${\displaystyle P_{1},P_{2}}$ with respect to ${\displaystyle {\tilde {L}}}$ are readily shown to be

${\displaystyle P_{1}={\frac {\partial {\tilde {L}}}{\partial {\dot {Q_{1}}}}}={\frac {\partial L}{\partial {\dot {Q_{1}}}}}+\lambda ={\frac {\partial L}{\partial {\dot {Q_{1}}}}}-{\frac {d}{dt}}{\frac {\partial L}{\partial {\dot {Q_{2}}}}}}$

${\displaystyle P_{2}={\frac {\partial {\tilde {L}}}{\partial {\dot {Q_{2}}}}}={\frac {\partial {L}}{\partial {\dot {Q_{2}}}}}}$

while

${\displaystyle P_{\lambda }=0}$

These are precisely the definitions given above by Ostrogradski. One may proceed further to evaluate the Hamiltonian

${\displaystyle {\tilde {H}}=P_{1}{\dot {Q_{1}}}+P_{2}{\dot {Q_{2}}}+p_{\lambda }{\dot {\lambda }}-{\tilde {L}}=P_{1}Q_{2}+P_{2}{\dot {Q_{2}}}-{L}}$,

where one makes use of the above Euler-Lagrangian equations for the second equality. We note that due to non-degeneracy, we can write ${\displaystyle {\ddot {q}}={\dot {Q_{2}}}}$ as ${\displaystyle a(Q_{1},Q_{2},P_{2})}$. Here, only three arguments are needed since the Lagrangian itself only has three free parameters. Therefore, the last expression only depends on ${\displaystyle P_{1},P_{2},Q_{1},Q_{2}}$, it effectively serves as the Hamiltonian of the original theory, namely,

${\displaystyle H=P_{1}Q_{2}+P_{2}a(Q_{1},Q_{2},P_{2})-L(Q_{1},Q_{2},P_{2})}$.

We now notice that the Hamiltonian is linear in ${\displaystyle P_{1}}$. This is a source of the 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.