generalized potential energies

We study particle dynamics under curl forces. These forces are a class of non-conservative, non-dissipative, position-dependent forces that cannot be expressed as the gradient of a potential function. We show that the fundamental quantity of particle dynamics under curl forces is a work 1-form. By using the Darboux classification of differential 1-forms on R² and R³, we establish that any curl force in two dimensions has at most two generalized potentials, while in three dimensions, it has at most three.