Trajectory equations for a non-conservative natural system

In practice, it is often necessary to know the trajectory of motion of natural mechanical systems. At present, the trajectory equations in configuration space are well known only for some conservative systems. It is also important to derive equations for systems in non-stationary external fields. In this paper, we prove a theorem on the change in kinetic energy, which states that the rate of kinetic energy change depends both on external forces and on the rate of metric tensor change. This theorem can be expressed geometrically as a combination of the products of forces and changes in the metric tensor with tangent vectors. Generalized velocities and accelerations are similarly described in terms of the tangent vectors and their derivatives along the trajectory. Substitution of these expressions into the Lagrange equations results in trajectory equations corresponding to the degrees of freedom of the system. The left-hand side contains a covariant derivative of the tangent vector, and the right-hand side includes a cubic polynomial of the tangent vectors. These equations represent the geometric form of the Lagrange equations, which can be solved numerically using the fourth order Runge-Kutta method. Together with the trajectory parameterization, these equations provide a trajectory method for solving dynamics problems.