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