From quantum dynamics of spin 1 particle in Coulomb field to jet geometric-physical objects M. Neagu, O. Florea, O. Veko, E. Ovsiyuk, V. Red’kov Abstract. The aim of this paper is to relate particular second order ordinary differential equations, associated with quantum mechanics of spin 1 particle in Coulomb field, to certain natural jet geometrical objects, such as a nonlinear connection, a distinguished (d-) torsion or a geometrical Yang-Mills stress-like construction. In its critical points the Yang-Mills entity has the value 1/4. This is intimately connected with the turning points of the quantity P 2 x , which is meaningful both in the context of classical mechanics and quantum mechanics. M.S.C. 2010: 34G10, 53B50, 53B99. Key words: homogeneous linear ODEs of second order; distinguished torsions; quantum mechanics; spin 1 particle in Coulomb field; Yang-Mills stress-like entity. 1 Jet geometrical objects produced by a homogeneous linear ODE of second order Starting from a given homogeneous linear ODE of superior order (generally of order n), in the monograph [1] it was constructed a natural collection of jet geometrical objects which geometrically characterize the initial ODE. More precisely, if we have the initial second order homogeneous linear ODE (1.1) d 2 Φ dr 2 + a 1 (r) dΦ dr + a 2 (r)Φ = 0, via the canonical ODEs system        dx 1 dr = x 2 := X (1) (1) (x 1 ,x 2 ) dx 2 dr = −a 2 (r)x 1 − a 1 (r)x 2 := X (2) (1) (x 1 ,x 2 ), where x 1 = Φ and x 2 = dΦ/dr, we can associate it with the following geometri- cal objects on the 1-jet space J 1 ([0, ∞), R 2 ), whose coordinates are (r, x 1 ,x 2 ,y 1 1 := dx 1 /dr, y 2 1 := dx 2 /dr) (for more details, please see [1, p. 175] or [4]): Applied Sciences, Vol. 16, 2014, pp. 72-98. c  Balkan Society of Geometers, Geometry Balkan Press 2014.