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

Sudip Garai, Partha Guha

Research output: Contribution to journalArticlepeer-review

3 Scopus citations

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 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.

Original languageBritish English
Pages (from-to)2257-2272
Number of pages16
JournalNonlinear Dynamics
Volume103
Issue number3
DOIs
StatePublished - Feb 2021

Keywords

  • Bertrand–Darboux
  • Complex curl force
  • Curl forces
  • Flapping and spinning saddle
  • Higher-order saddle potentials
  • Newton equation
  • Trap

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