Dynamic Equations of the Discrete Particle and Incompressible Continuous Medium (Field) for Generalized Coordinates System

Using the metric tensor to define the infinitesimal displacement vector and the kinetic energy, a generalized scalar function (Lagrangian) is given. It is inherently invariant under a change of coordinates. With this scalar function and the Euler-Lagrangian procedure for a conservative system, a generalized dynamic equation of motion in covariant vector form is derived: for the discrete particle model and for the continuous flow field model. The covariant acceleration vector is explicitly given. A great advantage of this expression is that it is easy to find the conservation law in the system and easy to reflect underlying geometric symmetries if the metric tensor is independent of a specific coordinate direction; an example of the spherical coordinate is shown in the paper. The metric tensor of the spherical coordinate is independent of the azimuthal (ϕ) coordinate. For the discrete particle model, the momentum exchanges between neighboring particles (forces due to the direct interactions between the particle and its neighbors) do not need to be considered; thereby, the kinetic energy is assumed to be independent of the coordinates to neglect the influence of neighboring particles. The velocity gradient (to be exact, the gradient of kinetic energy) is ignored. The equation degenerates to Newton’s second law for discrete particles, which can be located in a potential field. For a continuous flow field, the partial derivatives of the velocity with respect to the coordinates are included (the gradient of kinetic energy), so that the momentum exchanges between the particle and its neighbors are included (interactions between neighboring particles cause changes in kinetic energy and momentum). The kinetic energy gradient forms a product of a velocity gradient and velocity itself. This velocity gradient tensor can be decomposed into a symmetric part and an antisymmetric part to separate the local stretching or shrinking deformation (symmetric strain rate tensor) from the fluid parcel’s local rotation (antisymmetric). There is a fine difference between this model and the Navier-Stokes equations. The Navier-Stokes equations consider the momentum exchanges between fluid layers using a symmetric viscous stress tensor modeling, while this model is more general and adds an extra local rotational motion term to consider the interaction between velocity and vorticity, exactly to say, adding a half of the cross product of the velocity and vorticity vectors. This analysis primarily relies on a pure theoretical framework without direct validation against experimental data; further work should be conducted for fine experimental validations to support the theoretical findings. It should be emphasized that this model is only valid for a conservative system, namely without considering the irreversible process, such as viscous dissipations.