The general relativistic perturbations of a uniformly rotating and axisymmetric elastic deformable astronomical body in the first post-Newtonian approximation of Einstein's general relativity is discussed in a co-rotating coordinates. The main new results presented here are the post-Newtonian variations of the energy and Euler equation, which at the Newtonian level of accuracy is so called Jeffreys-Vicente equation and it play a fundamental role for the description of global geodynamics in the classical geophysics. Our results will be useful to treat the relativistic nutation and precession of the Earth and other planets.