Stud. Univ. Babe¸ s-Bolyai Math. 59(2014), No. 3, 385–391 On the geometry of conformal Hamiltonian of the time-dependent coupled harmonic oscillators Hengameh Raeisi-Dehkordi and Mircea Neagu Abstract. In this paper we construct the distinguished (d-) geometry (in the sense of d-connections, d-torsions, d-curvatures, momentum geometrical gravita- tional and electromagnetic theories) for the conformal Hamiltonian of the time- dependent coupled oscillators on the dual 1-jet space J 1* ( R, R 2 ) . Mathematics Subject Classification (2010): 70S05, 53C07, 53C80. Keywords: Conformal Hamiltonian of time-dependent coupled harmonic oscilla- tors, Cartan canonical connection, d-torsions, d-curvatures, geometrical Einstein- like equations. We dedicate this paper to the memory of Professor Gheorghe Atanasiu (1939-2014). 1. Introduction The model of time-dependent coupled oscillators is used to investigate the dy- namics of charged particle motion in the presence of time-varying magnetic fields. At the same time, the model of coupled harmonic oscillator has also been widely used to study the quantum effects in mesoscopic coupled electric circuits. For more details, please see [2]. If m i (i =1, 2), ω i (i =1, 2), and k(t) are the time-dependent mass, frequency, and the coupling parameter, respectively, then the conformal Hamiltonian of the time- dependent coupled harmonic oscillators is given on the dual 1-jet space J 1* ( R, R 2 ) by [2] H(t, x, p)= h 11 (t)e σ(x) " ( p 1 1 ) 2 m 1 (t) + ( p 1 2 ) 2 m 2 (t) # + F (t, x)= (1.1) = h 11 (t)e σ(x) δ ij m i (t) p 1 i p 1 j + F (t, x)= = h 11 (t)g ij (t, x)p 1 i p 1 j + F (t, x),