Geometrical Particle Models In Lorentz Space L₄ On Non-null Curves

In this paper we study the particle models in the Lorentz space L₄ on the non-nullcurves.We consider the functional:∫_rf(k₁,k₂)ds is a smooth real function ofk₁ and k₂.In order to solve the corresponding Euler-Lagrange equations,three Killing vectorfields:P₁,P₂,P₃,where P₁,P₂ are translational vector fields and P₃ is rotational one,we willobtain three constants c,which are also first integral constants of Euler-Lagrange equation.Then we decrease the degrees of freedom of the Euler-Lagrange equation.We also used therelations among P₁,P₂,P₃ to solve the motion equations.If we set the planeΠto be spannedby P₁ and P₂.Furthermore in two cases:Πis degenerate andΠis non-degenerate,we givethe corresponding represents of the motion equations by cylindrical coordinates.We expressthe first curvature k₁ of the critical curves by using Jacobi elliptical functions for the elasticfunctional∫_rk₁²ds.For the elastic functional∫_rak₁+bk₂ds,where a,b are any constants inR,we give the relation of the curvatures:k₁,k₂,k₃ of the respective critical curves.