Verifications for Multiple Solutions of Triaxial Earth Rotation

Abstract: In this study, we provide several verifications on existence for multiple solutions of rotation with triaxial Earth. In fact, the exact solution for Euler rotation equations of triaxial Earth case appears as elliptic functions of double periods in which the main term is Chandler wobble and the other is a periodic polar motion of 14.6 year. By quadratic polynomial approach, the triaxial condition of the principal moments of inertia satisfying inequality A<B<C simultaneously guarantees that two stable solutions exist. On the other hand, prime algorithm provides algebraic equations of two elliptic curves and a hyperbolic trajectory simultaneously so that shows the same consequences. As standard deduction for nonlinear differential equations, Jacobian matrix and its eigen values show that there are two stable and an unstable solutions for the rigid body rotation model with true solutions projection onto three principal axes. By observation data series of polar motion and LOD, we also provide filtering results illustrating explicit period of about 14 years oscillation. At last, phase portrait of the rotation equations without solving process is drawn with illustration of 2-torous graph in X—Y plane with spirals shows clearly existence of two periods and other two projections of phase portrait appear as explicit trajectory of single pendulum with hyperbolic trajectories in it illustrating complex permanent consequence in multiple solutions of tentative trajectories