Mei symmetry and conserved quantities in Kirchhoff thin elastic rod statics

Abstract

We investigate the application of the Mei symmetry analysis in finding conserved quantities for the thin elastic rod statics. By using the Mei symmetry analysis, we have obtained the Jacobi integral and the cyclic integrals for a thin elastic rod with intrinsic twisting for both the cases of circular and non-circular cross sections. Our results can be easily reduced to the results without the intrinsic twisting that have been reported. Through calculation, we find that the Noether symmetry can be more directly and easily used than the Mei symmetry in finding the first integrals for the thin elastic rod. These first integrals will be helpful in the study of exact solutions and stability, as well as the numerical simulation of the elastic rod model for DNA.

Conformal invariance of Mei symmetry and conserved quantities of Lagrange equation of thin elastic rod

“…For thin elastic rod with circular cross section, the twist torque of cross section is invariant along the arc coordinate; that is, there exist cyclic integrals [57] C q 1 cosq 2 + q 3 = const (28) When the thin elastic rod is in equilibrium and has no external force or torque imposed on the rod except the two ends, the internal principal vector is invariant [60]. We suppose F = 0.…”

“…Xue et al [56] studied the conserved quantities in general theorems of elastic rod dynamics. [57,58] studied the Mei symmetry and conserved quantities of Kirchhoff elastic rod statics and Noether theorem of Cosserat elastic rod dynamics.…”

Noether symmetry and conserved quantities of the analytical dynamics of a Cosserat thin elastic rod

“…As α = α 1 , considering Eqs. ( 12) and ( 13), equation (30) denotes the generalized internal force on the two ends of rod, and equation (31) denotes the total momentum of a rod conserved along the arc length. An elastic rod under the action of a equilibrant couple on the two ends [44] coincides with the condition (30), so it has conserved quantities (31).…”

“…[9] The symmetry method has been developed as a general method for solving the differential equations of mechanical systems. In addition to Noether symmetry, [9][10][11][12][13][14][15][16][17][18] Lie symmetry for mechanical systems has also been developed, [19][20][21][22][23][24][25][26][27] as well as Mei symmetry (form invariance), [28][29][30][31] unified symmetry, [32,33] conformal invariance, [34][35][36][37] and other symmetries. [38,39] The application of symmetry methods in mechanical systems is referred to several studies.…”

Conformal Invariance and Conserved Quantities for Lagrange Equation of Thin Elastic Rod

“…In Refs. [42,43] the authors studied the Mei symmetry and conserved quantities of Kirchhoff elastic rod statics and Noether theorem of Cosserat elastic rod dynamics. In Ref.…”