A Note on Reparametrizations of the Euler Equations

Analytic and numeric comparisons of the solution of Euler equations for the free rigid body problem are accomplished, after applying two linearizations. The one proposed by Molero et al. makes use of a solution in trigonometric functions whereas the other one comes as an application of a recent theorem by Llibre et al. and expresses it in exponential functions as a result of a complete linearization of the equations. Both approaches avoid the use of the Jacobi elliptic functions, whose computations are slower. Only an elliptic integral of the first kind remains in the process to get the original physical time where positions of the solution trajectory are taking place, and the overall process is less time consuming than the elliptic functions. Conclusions show that, depending on the requirements of each application, a combination of both approaches is a very competitive solution as an alternative to elliptic functions.