The Darboux Classification of Curl Forces

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 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 $\mathbb{R}^2$ and $\mathbb{R}^3$, we establish that any curl force in two dimensions has at most two generalized potentials, while in three dimensions, it has at most three. These potentials generalize the single potential of conservative systems. For any curl force field, we introduce a corresponding conservative force field -- the conservative auxiliary force. The Hamiltonian of this conservative force is a conserved quantity of motion for the dynamics of a particle under the curl force, although it is not the physical energy.