The First Variational Formula and the Ostrogradsky Formalism

Abstract

We present a derivation at a level suitable for undergraduates of the Ostrogradsky formalism for Lagrangians in classical mechanics that depend upon an arbitrary number of time derivatives of the configuration. From the boundary term in the first variation of the Lagrangian we derive the Ostrogradsky formulas that define the Hamiltonian formulation of mechanical systems. Worked examples, exercises, and applications to the literature are also provided. An accompanying computer program that implements the formalism is discussed in the Supplementary Materials, and code for computing Hamiltonians via the Ostrogradsky formalism is provided in the Supplementary Materials and in a GitHub repository.