Energy function of 2D and 3D coarse systems

In this work, while using the Flow Curvature Method developed by one of us (JMG), we prove that the energy function of two and three-dimensional coarse systems involving a small parameter $\mu$ can be directly deduced from the curvature of their trajectory curves when $\mu$ tends to zero. Such a result thus confirms the relationship between curvature and energy function for a certain class of differential systems already established in one of our previous contributions. Then, we state that the rate of change of the energy function of such coarse systems is equal to the scalar product of the velocity vector field and its first time derivative, i.e. the acceleration vector field. The comparison of these results with the so-called Frénet frame enables to prove that energy function is proportional to the normal component of the acceleration when $\mu$ tends to zero while the rate of change of the energy function is proportional to the tangential component of the acceleration at first order in $\mu$. Two and three-dimensional examples are then used to emphasize these two main results.