Symmetry analysis of the constant acceleration curve equation

Abstract

Abstract Lie group theory is applied to the curve equation which maintains constant normal accelerations for a vehicle with constant deceleration. The curve equation is a third order nonlinear ordinary differential equation for which the symmetries are calculated. It is shown that the equation possesses four-parameter Lie group of transformations including scaling, rotation and translational symmetries. In the case of constant velocity, the algebra increases to a six-parameter Lie group of transformations. Using the symmetries of the differential equation, the group invariant solutions are determined first. The conditions for group invariant solutions to exist are given. By employment of the symmetries, a solution is obtained by reduction of order also. It is found that the nontrivial solutions are of implicit complex forms.