Intrinsic variational structure of higher-derivative formulations of classical mechanics

Abstract

This paper investigates the geometric structure of higher-derivative formulations of classical mechanics. It is shown that every even-order formulation of classical mechanics higher than the second order is intrinsically variational, in the sense that the equations of motion are always derivable from a minimum action principle, even when the system is non-Hamiltonian. Particular emphasis is placed on the fourth-order formulation, as that is shown to be the lowest order for which the governing equations are intrinsically variational. The Noether symmetries and associated conservation laws of the fourth-order formulation, including its Hamiltonian, are derived along with the natural auxiliary conditions. The intrinsic variational structure of higher-derivative formulations makes it possible to treat non-Hamiltonian systems as if they were Hamiltonian, with immediate classical applications. A case study of the classical damped harmonic oscillator is presented for illustration, and an action is formulated for a higher-order Navier-Stokes equation.