Circular Cross Section Of The Super - Thin Elastic Rod Model Noether Symmetry, Lie Symmetry And Conserved Quantity,

Many slender mechanics,such as electric cables,ropes,textile fibers,can be modeled by super-long elastic rod,which have been extensively studied in the last centuries.There have been many results published in important articles and books.Since the double helix model brought out by Watson and Crick in 1953,many breakthroughs have been made on DNA's modeling,theoretical analysis and experimental techniques, which greatly promoted the development of genetic engineering.One special characteristic of DNA is that its diameter,compare with its length,is tens of million times shorter,which can not be described by classical models of an elastic rod, so it is called super-long elastic rod by many authors.An important model to describe a super-long elastic rod is based on the Kirchhoff's dynamic analogue technique,by which we treat the curve length s as a quasi-time and the elastic rod as a rigid cross section moving evenly along the radian coordinate of the centerline with 'time's.These lead to a structural model of the super slender rod is similar to the dynamical model of a rigid body,which means the classic results of dynamical system can be used to analyze the structural properties of the super-long elastic rod,and the conserved quantities can be obtained by using mathematical methods of dynamical systems such as Noether symmetries and Lie symmetries.In this paper the Kirchhoff's technique is used as a bridge between particle dynamics and structural mechanics of super long elastic rod.Based on this idea,we apply Noether theory and Lie group method to analyze the symmetric properties of super long elastic rod.By introducing Hamilton canonical equations and applying infinitesimal transformation corresponding to radian coordinates,generalized coordinates and generalized momentum,we derived the Noether symmetries,Lie symmetries and the conserved quantities.