Lorentz-type equations in first-order jet spaces endowed with nonlinear connection Vladimir Balan Abstract. Sections 1 and 2 describe the notions of nonlinear and N -linear connec- tion in first order jet spaces J 1 (T,M), point out relevant subclasses which include the Cartan and Berwald connections and remind the 1-jet Lagrangian case for elec- tromagnetism. In section 3 are derived the associated electromagnetic tensors and the corresponding generalized Lorentz equations. It is shown that in the unipara- metric case these equations confine to the already known ones of the tangent bundle framework. Section 4 exemplifies by numerical simulation of the obtained Lorentz equations, the deformation of geodesics to Lorentz curves as effect of the electromag- netic field influence. Mathematics Subject Classification 2000: 58A20, 58A30, 53B15, 53B40, 53B50, 53C60, 53C80, 53C22, 78A35. Key words: jet space, Cartan connection, Berwald connection, nonlinear connec- tion, deflection tensor, electromagnetic tensor, Lorentz equations, stationary curves, geodesics, graphical simulation. 1. Nonlinear connections on J 1 (T,M ) Let ξ =(E = J 1 (T,M ),π,T × M ) be the first order jet bundle of mappings ϕ : T → M , where T and M are C ∞ real differentiable manifolds with dim T = m, dim M = n respectively. We shall denote the local coordinates in E by (t α ,x i ,y A ) (α,i,A)∈I* ≡ (y μ ) μ∈I , where we use the notations I * = I h1 × I h 2 × I v , I h1 = 1, m, I h 2 = m +1,m + n, I v = m + n +1,m + n + mn, I h = I h1 ∪ I h 2 ,I = I h ∪ I v . The indices will implicitly take values as follows: α,β,... ∈ I h 1 ; i, j, . . . ∈ I h 2 ; A,B,... ∈ I v ; λ, μ, . . . ∈ I. When appropriate, for A = m + n + n(i - m - 1) + α, we identify A ≡ ( i α ) and y A ≡ x ( i α ) = ∂x i ∂t α . A non-linear connection N = {N A μ } μ∈I h ,A∈Iv on E provides a splitting [3] TE = HE ⊕ V E, (1) Proceedings of The First French-Romanian Colloquium of Numerical Physics, October 30-31, 2000, Bucharest, Romania. c  Geometry Balkan Press 2002. The present contributed work was partially supported by Grant CNCSIS MEN 34967 (675) / 2001 and by Macedonian Grant 08-2076 (4) / 2001.