Invariant linearization criteria for a three-dimensional dynamical system of second-order ordinary differential equations and applications

• Using the projection method, we project the system of geodesic equations down by one dimension and write the system of second order Ordinary differential equations (ODEs).
• The linearization criteria for the corresponding system of geodesic equations are imposed by means of the vanishing of the Riemann tensor.
• A necessary and sufficient condition for the system of three cubically semi-linear ODEs to be linearizable via a point transformation to the simplest system of three second-order ODEs is given as a theorem.
• The linearization criteria for the three quadratically semi-linear ODEs and ODEs linear in the first derivatives are also given.
• A few applications, including those taken from classical mechanics, are given.

Second-order dynamical systems are of paramount importance as they arise in mechanics and many applications. It is essential to have workable explicit criteria in terms of the coefficients of the equations to effect reduction and solutions for such types of equations. One important aspect is linearization by invertible point transformations which enables one to reduce a non-linear system to a linear system. The solution of the linear system allows one to solve the non-linear system by use of the inverse of the point transformation. It was proved that the n-dimensional system of second-order ordinary differential equations obtained by projecting down the system of geodesics of a flat (n+1)-dimensional space can be converted to linear form by a point transformation. This is a generalization of the Lie linearization criteria for a scalar second-order equation. In this case it is of the maximally symmetric class for a system and the linearizing transformation as well as the solution can be directly written down. This was explicitly used for two-dimensional dynamical systems. The criteria were written down in terms of the coefficients and the linearizing transformation allowed for the general solution of the original system. Here the work is extended to a three-dimensional dynamical system and we find explicit criteria, including the linearization test given in terms of the coefficients of the cubic in the first derivatives of the system and the construction of the transformations, that result in linearization. Applications to equations of classical mechanics and relativity are given to illustrate our results.