First-order equations of motion for classical mechanics

Abstract

A global canonical first-order equation of motion is derived for any mechanical system obeying Newton’s second law. The existence of a Lagrangian is not assumed, but the properties of the canonical equation are similar to those of the Hamiltonian formulation. The choice of map F from velocity space to phase space is not determined by the condition that the first-order equation of motion be equivalent to a second-order equation on configuration space and therefore is left open to be selected on the basis of other considerations. The canonical equation is a covector or 1-form equation on the Whitney sum T*Q⊕TQ and contains the second-order equation condition, restriction to the graph of the map F, and Newton’s equation of motion in first-order form. The last is related to Newton’s second-order equations by the consistency condition that the motions not lead off graph F in T*Q⊕TQ. The first-order equation of motion can be projected onto phase space if the map F can be inverted.