A Software Implementation of Ostrogradsky’s Construction for Higher Order Lagrangians

The Lagrangian and Hamiltonian formulations of classical mechanics are powerful tools for characterizing the behavior of physical systems. The process to transform a Lagrangian depending on one time derivative to a corresponding Hamiltonian via a Legendre transform is well known. The Ostrogradsky construction generalizes Hamilton’s equations to account for an arbitrary number of time derivatives in the Lagrangian. We will present this construction along with a software implementation in Maple. Interestingly, these Hamiltonians are known to exhibit pathological behavior in some cases—most notably possessing energies unbounded from below (known as Ostrogradsky instability)—and we explore this feature.