Higher-order saddle potentials, nonlinear curl forces, trapping and dynamics

The position-dependent non-conservative forces are called curl forces introduced recently by Berry and Shukla (J Phys A 45:305201, 2012). The aim of this paper is to study mainly the curl force dynamics of non-conservative central force x¨ = - xg(x, y) and y¨ = - yg(x, y) connected to higher-order saddle potentials. In particular, we study the dynamics of the type x¨i=-xig(12(x12-x22)), i= 1 , 2 and its application towards the trapping of ions. We also study the higher-order saddle surfaces, using the pair of higher-order saddle surfaces and rotated saddle surfaces by constructing a generalized rotating shaft equation. The complex curl force can also be constructed by using this pair. By the direct computation, we show that all these motions of higher-order saddles are completely integrable due to the existence of two conserved quantities, viz. energy function and the Fradkin tensor. The Newtonian system x¨ = X(x, y) , y¨ = Y(x, y) has also been studied with an energy like first integral I(x,x˙)=12x˙TM(x)x˙+U(x), where M(x) is a (2 × 2) matrix of which the components are polynomials of degree less than or equal to two and the condition on X and Y for which the curl is non-vanishing is also obtained.