Parabolic Cylindrical Description of Velocity and Acceleration Tensor in Bipolar Coordinate Using Tensor Analysis

Instantaneous velocity and acceleration are often studied and expressed in Cartesian, Circular Cylindrical, and Spherical Coordinate systems but it is a well-known fact that some bodies cannot be perfectly described in these coordinate systems, so they required some other curvilinear systems such as oblate spheroidal, parabolic cylindrical, elliptic cylindrical bipolar, and others. It is of important interest in theoretical physics to establish equations of motion in Bipolar Coordinate system, which is essentially suitable to accurately describe the motion of bipolar bodies such as hyperbolas, ellipses and other curves in the universe. In this work, we derived a new expression for the instantaneous velocity and acceleration vector of bodies and test particles in the bipolar coordinate system using tensor analysis. The Laplacian of a scalar field in this bipolar coordinate system was also derived which has applications in mechanics.