Hamiltonian description of nonlinear curl forces from cofactor systems

Recently, Berry and Shukla presented (J Phys A 45:305201, 2012; J Phys A 46:422001, 2013; Proc R Soc A 471:20150002, 2015) a fundamental new dynamics concerning forces (accelerations) depending only on position, i.e. without velocity-dependent dissipation, which was partly anticipated in the papers of cofactor systems introduced by the Linköping school (Rauch-Wojciechowski et al. in J Math Phys 40:6366–6398, 1999; Lundmark in Stud Appl Math 110(3):257–296, 2003; Lundmark in Integrable nonconservative Newton systems with quadratic integrals of motion. Linköping Studies in Science and Technology. Thesis No. 756, Linköping Univ., Linköping, 1999). In this paper, we extend their results to nonlinear curl forces, where the nonlinearity is with respect to the coordinate dependence of the forces, and study the Hamiltonians for homogeneous quadratic and cubic cases presenting the conditions for existence of Hamiltonian curl forces. In particular, we examine the existence and expressions of the Hamiltonian curl forces for planar systems when the accelerations are given by general (both homogeneous and inhomogeneous) second-order and homogeneous cubic polynomials, and also associate the cubic case with an example from optics.