On the class of two-dimensional geodesic curves in the field of the gravity force

In this paper, without using methods of variational calculus, the problem of finding a geodesic in a curved space with respect to gravitational and dissipative forces was solved. Solving it, we use the most convenient polar coordinates r,ф . The basic assumption relies on the fact that dynamical motion equations written in curvilinear coordinates in which the Riemann curvature R is different from zero rather strongly differ from similar equations in the case of a flat space. To obtain the required equation of a geodesic arc, a contravariant vector of the velocity d_xⁱ v^l =- was introduced. For this vector, with regard to all active forces, the following equation dt was solved: dvⁱ_k , _i F' -T + n,v^kv' = gⁱ +- , dt m gⁱ are acceleration components of the gravitational force of the two-dimensional r-ф space, and the dissipative force is F' = k₁^ikN^k + k₂vⁱ, k₁^ik are tensor components of the dry friction, k₂ is the coefficient of the viscous friction, and N ⁱ are the force components. Provided that the scalar curvature of Riemann is different from zero, дГ дГ^ф₂ r = фф___^гф + Гr ГФ _- ГФ Гr = - Ф 0 d_r d_r фф Ф^г Ф^г фф _r 2 ' a nonlinear system of differential equations governing the required geodesic was obtained in the polar coordinates r and ф: _F r - rep² = g sin ф + -cos (а -ф), m _F rep + 3np = - g cos ф --sin (а - ф), m where r = |r| = |ix + jy| is the length of the radius-vector drawn from the origin of coordinates to the observation point M lying on the geodesic line y = y(x), ф is the polar angle of the reference point, and a is the acute angle between the tangent drawn to the point of M and to axis of abscissas. The analytical and numerical solutions of this system in the absence of the resistance forces, i.e. Fj_r = 0 , showed the great difference between the found geodesic and the parabola typical for the case of free fall of bodies in the gravitational field in Euclidean space. AMS Mathematical Subject Classification: 53Z05

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