Inapplied mathematics, theOstrogradsky instability is a feature of some solutions of theories havingequations of motion with more than two timederivatives (higher-derivative theories). It is suggested by a theorem ofMikhail Ostrogradsky inclassical mechanics according to which a non-degenerateLagrangian dependent ontime derivatives higher than the first corresponds to aHamiltonianunbounded from below. As usual, the Hamiltonian is associated with the Lagrangian via aLegendre 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.^[1]However, Ostrogradsky's theorem does not imply that all solutions of higher-derivative theories are unstable as many counterexamples are known.^{[2][3][4][5][6][7][8][9][10]}

Source:^[11]

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

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

Non-degeneracy of${\displaystyle L}$ means that thecanonical coordinates can be expressed in terms of the derivatives of${\displaystyle q}$ and vice versa. Thus,${\displaystyle \mathrm{\partial }L/\mathrm{\partial }\ddot{q}}$ is a function of${\displaystyle \ddot{q}}$ (if it were not, theJacobian${\displaystyle det[{\mathrm{\partial }}^{2}L/(\mathrm{\partial }{\ddot{q}}_i\mathrm{\partial }{\ddot{q}}_j)]}$ 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,

${\displaystyle P_1:=\frac{\mathrm{\partial }L}{\mathrm{\partial }\dot{q}}-\frac{d}{dt}\frac{\mathrm{\partial }L}{\mathrm{\partial }\ddot{q}}}$

${\displaystyle P_2:=\frac{\mathrm{\partial }L}{\mathrm{\partial }\ddot{q}}}$

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

${\displaystyle Q_1:\frac{d}{dt}\frac{\mathrm{\partial }L}{\mathrm{\partial }\dot{Q_1}}+\dot{\lambda }-\frac{\mathrm{\partial }L}{\mathrm{\partial }{Q}_1}=0}$,

${\displaystyle Q_2:\frac{d}{dt}\frac{\mathrm{\partial }L}{\mathrm{\partial }\dot{Q_2}}+\lambda =0}$,

${\displaystyle \lambda :Q_2-\dot{Q_1}=0}$,

Now, the canonical momentum${\displaystyle P_1,P_2}$ with respect to${\displaystyle \tilde{L}}$ are readily shown to be

${\displaystyle P_1=\frac{\mathrm{\partial }\tilde{L}}{\mathrm{\partial }\dot{Q_1}}=\frac{\mathrm{\partial }L}{\mathrm{\partial }\dot{Q_1}}+\lambda =\frac{\mathrm{\partial }L}{\mathrm{\partial }\dot{Q_1}}-\frac{d}{dt}\frac{\mathrm{\partial }L}{\mathrm{\partial }\dot{Q_2}}}$

${\displaystyle P_2=\frac{\mathrm{\partial }\tilde{L}}{\mathrm{\partial }\dot{Q_2}}=\frac{\mathrm{\partial }L}{\mathrm{\partial }\dot{Q_2}}}$

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, onlythree 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 theoriginal 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${\displaystyle P_1}$ and hence unbounded from below. 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.

• Lagrangian mechanics
• Hamiltonian mechanics
• Calculus of variations
• Mathematical physics

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