The Theorem of Ostrogradsky

TL;DR

Ostrogradsky's theorem shows that nondegenerate Lagrangians with higher time derivatives yield Hamiltonians that are linear in a subset of canonical momenta, making the theory generically unstable and energetically unbounded. Woodard surveys the canonical construction, the resulting instability, and the challenges posed by quantization and degeneracy, arguing that simple evasion strategies fail in realistic 3+1D interacting field theories. He explains the physical consequences, including vacuum decay into positive- and negative-energy sectors and the nondecoupling of high-momentum modes, highlighting why higher-derivative counterterms or nonlocal models struggle as fundamental descriptions. The conclusion emphasizes Ostrogradsky’s theorem as a foundational restriction on gravity modifications and other local theories, while noting generalizations to fermionic systems and the persistent difficulty of constructing stable, interacting, higher-derivative theories.

Abstract

Ostrogradsky's construction of a Hamiltonian formalism for nondegenerate higher derivative Lagrangians is reviewed. The resulting instability imposes by far the most powerful restriction on fundamental, interacting, continuum Lagrangian field theories. A discussion is given of the problems raised by attempts to evade this restriction.