Geometrical Particle Models In 4-D Lorentz Space Forms On Non-null Curves

In this paper, we study the particle models of the non-null curves in 4-dimensional Lorentz space forms. For any functional:∫γf(K1,K2)ds, where f(K1,k2) is a smooth real function depending on the first curvature k1 and the second curvature k2, We obtain the Euler-Lagrange motion equations of the relativistic particle model and construct a Killing vector field which along the critical curve. Then we get two Killing vector fields in 3-dimensional Lorentz space forms and two first integrals respectively. At last, we ob-tain 9 parametric equations of the critical curve according to the 9 representative elements of o(4,2).