Third order equations of motion and the Ostrogradsky instability

TL;DR

This paper demonstrates that third order equations of motion in analytical mechanics inherently suffer from the Ostrogradsky instability unless degeneracy conditions are met, extending the analysis to higher odd orders.

Contribution

It proves that nondegenerate third order Lagrangians inevitably lead to instability, clarifying limitations of higher order equations in classical mechanics.

Findings

Third order equations exhibit Ostrogradsky instability without degeneracy.

Degeneracy conditions are necessary to avoid instability in higher order Lagrangians.

Extension of instability analysis to higher odd order equations.

Abstract

It is known that any nondegenerate Lagrangian containing time derivative terms higher than first order suffers from the Ostrogradsky instability, pathological excitation of positive and negative energy degrees of freedom. We show that, within the framework of analytical mechanics of point particles, any Lagrangian for third order equations of motion, which evades the nondegeneracy condition, still leads to the Ostrogradsky instability. Extension to the case of higher odd order equations of motion is also considered.