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

Abstract

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 $\ddot{x} = -xg(x,y)$ and $\ddot{y} = -yg(x,y)$ connected to higher-order saddle potentials. In particular, we study the dynamics of the type $\ddot{x}_i = -x_ig \big (\frac{1}{2}(x_{1}^{2} - x_{2}^{2}) \big )$, $i=1,2$ 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 $\ddot{x} = {{\mathcal {X}}}(x,y)$, $\ddot{y} = {{\mathcal {Y}}}(x,y)$ has also been studied with an energy like first integral $I(\mathbf{x}, \dot{\mathbf{x}}) = \frac{1}{2}\dot{\mathbf{x}}^TM(\mathbf{x})\dot{\mathbf{x}} + U(\mathbf{x})$, where $M(\mathbf{x})$ is a $(2 \times 2)$ matrix of which the components are polynomials of degree less than or equal to two and the condition on ${{\mathcal {X}}}$ and ${{\mathcal {Y}}}$ for which the curl is non-vanishing is also obtained.