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

S. O. Gladkov ; S. B. Bogdanova

Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 58 (2019), pp. 5-13

Résumé

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 $r, \varphi$. 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 $\nu^i=\frac{dx^i}{dt}$ was introduced. For this vector, with regard to all active forces, the following equation was solved: $$ \frac{d\nu^i}{dt}+\Gamma_{kl}^i\nu^k\nu^l=g^i+\frac{F^i}m, $$ $g^i$ are acceleration components of the gravitational force of the two-dimensional $r-\varphi$ space, and the dissipative force is $$ F^i=k_1^{ik}N^k+k_2\nu^i, $$ $k_1^{ik}$ are tensor components of the dry friction, $k_2$ 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, $$ R=\frac{\partial\Gamma^r_{\varphi\varphi}}{\partial r}-\frac{\partial\Gamma^{\varphi}_{r\varphi}}{\partial r}+\Gamma^r_{\varphi\varphi}\Gamma^{\varphi}_{r\varphi}-\Gamma^{\varphi}_{r\varphi}\Gamma^r_{\varphi\varphi}=\frac2{r^2}\ne0, $$ a nonlinear system of differential equations governing the required geodesic was obtained in the polar coordinates $r$ and $\varphi$: $$ \begin{cases} \ddot{r}-r\dot{\varphi}^2=g\sin\varphi+\frac{F_{\mathrm{fr}}}{m}\cos(\alpha-\varphi),\\ r\ddot{\varphi}+3\dot{r}\dot{\varphi}=-g\cos\varphi-\frac{F_{\mathrm{fr}}}{m}\sin(\alpha-\varphi), \end{cases} $$ where $r=|\mathbf{r}|=|\mathbf{i}x+\mathbf{j}y|$ is the length of the radius-vector drawn from the origin of coordinates to the observation point $M$ lying on the geodesic line $y=y(x)$, $\varphi$ is the polar angle of the reference point, and $\alpha$ 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. $F_{fr}=0$, 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.

Détail

Keywords: geodesic, Riemann tensor
Mots-clés : dynamical equations.