Nonrelativistic geodesic motion (Q1977395)

The authors show that any second-order dynamic equation on a configuration space \(X\to\mathbb{R}\) of nonrelativistic time-dependent mechanics can be seen as a geodesic equation with respect to a nonlinear connection on the tangent bundle \(TX\to X\) of relativistic velocities. The key point in this paper is the following statement. Let \(J^2X\) be the second-order jet manifold of \(X\to \mathbb{R}\), coordinated by \((x^\lambda\), \(x^i_0\), \(x^i_{00})\). Then any second-order dynamic equation \(x^i_{00}=\xi^i\) \((x^0,x^j,x^j_0)\) of nonrelativistic mechanics on \(X\to \mathbb{R}\) is equivalent to the geodesic equation \(\ddot x^0=0\), \(\dot x^0 = 1\), \(\ddot x^i = \overline K^i_j\dot x^0+\overline K^i_j\dot x^j\) with respect to a connection \(\overline K\) on \(TX\to X\) which fulfills the conditions \(\overline K^0_\lambda = 0\), \(\xi^i = \overline K^i_0+x^i_0\overline K^i_j|_{\dot x^0=1,\;\dot x^i=x^i_0}\). Relativistic and nonrelativistic geodesic equations are compared. It is shown that both relativistic and nonrelativistic equations of motion can be considered as the geodesic equations on the same tangent bundle \(TX\). The authors also investigate the Jacobi vector fields along nonrelativistic geodesics.