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
Mikhail Vasilyevich Ostrogradsky (transcribed also ''Ostrogradskiy'', Ostrogradskiĭ) (russian: Михаи́л Васи́льевич Острогра́дский, ua, Миха́йло Васи́льович Острогра́дський; 24 Sep ...
in
classical mechanics
Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classi ...
according to which a non-degenerate
Lagrangian
Lagrangian may refer to:
Mathematics
* Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier
** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
dependent on time derivatives higher than the first corresponds to a
Hamiltonian
Hamiltonian may refer to:
* Hamiltonian mechanics, a function that represents the total energy of a system
* Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system
** Dyall Hamiltonian, a modified Hamiltonian ...
unbounded from below. As usual, the
Hamiltonian
Hamiltonian may refer to:
* Hamiltonian mechanics, a function that represents the total energy of a system
* Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system
** Dyall Hamiltonian, a modified Hamiltonian ...
is associated with the Lagrangian via a
Legendre transform
In mathematics, the Legendre transformation (or Legendre transform), named after Adrien-Marie Legendre, is an involutive transformation on real-valued convex functions of one real variable. In physical problems, it is used to convert functions ...
. 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
. The
Euler–Lagrange equation
In the calculus of variations and classical mechanics, the Euler–Lagrange equations are a system of second-order ordinary differential equations whose solutions are stationary points of the given action functional. The equations were discovered ...
is
:
Non-degeneracy of
means that the
canonical coordinates can be expressed in terms of the derivatives of
and vice versa. Thus,
is a function of
(if it was not, the
Jacobian