Problem of Metrizability for the Dynamical Systems Accepting the Normal Shift.pdf

PROBLEM OF METRIZABILITY FOR THE DYNAMICAL SYSTEMS ACCEPTING THE NORMAL SHIFT. Sharipov R.A. 3 9 June, 1993. 9 1 Abstract. The problem of metrizability for the dynamical systems accepting the normal shift is formulated and solved. The r ¨ ˙ p explicit formula for the force field of metrizable Newtonian dynamical system r = F(r, r) is found. A 8 1 1. Introduction. 1 v The class of Newtonian dynamical systems accepting the normal shift was first defined in [1] (see also [2] and [3]). 3 It’s the class of dynamical systems in Rn given by the differential equations of the form 0 0 ¨ ˙ 4 (1.1) r = F(r, r) 0 4 and possessing some additional geometrical property: the property of conserving the orthogonality of trajectories 9 and the hypersurfaces shifted along these trajectories. The idea for considering such systems was found as a result / t of generalizing the classical construction of normal shift which is known also as a Bonnet transformation. Let S be n n ′ i the hypersurface in R . From each point M on S we draw the segment MM with the fixed length l along the - ′ ′ ′