Are nonlocal Lagrangian systems fatally unstable?
Carlos Heredia, Josep Llosa
TL;DR
The paper challenges the widespread claim that higher-derivative and genuinely nonlocal Lagrangians are inherently unstable by applying Lyapunov stability with suitable integrals of motion. It analyzes the Pais-Uhlenbeck oscillator in free, externally coupled, internally coupled, and ghost-augmented forms, plus a fully nonlocal infinite-order model, to derive precise conditions under which stability persists even when the Hamiltonian is not bounded below. By leveraging the direct Lyapunov method and carefully constructed invariants, the authors show that many systems exhibit Lyapunov stability across nontrivial couplings, thereby clarifying misconceptions about Ostrogradsky instability. The work provides practical benchmarks for designing stable higher-derivative and nonlocal theories with potential cosmological and field-theoretic relevance.
Abstract
We prove that higher-derivative and genuinely nonlocal Lagrangian systems can be Lyapunov-stable even when their Hamiltonians lack a lower bound. Explicit free and coupled Pais-Uhlenbeck oscillators, together with a genuine nonlocal model, are analysed to identify the precise conditions under which stability holds. These counterexamples point out the logical gap in the "Ostrogradsky instability" claims and provide benchmarks for constructing efficient stable higher-derivative theories.