On The Precession Of Planets With Action-angle Variables And General M(?)ller Transformation

It is well known that, general relativity is the theory of space, time, and gravitation formulated by Einstein in 1915. In this dissertation, based on the Einstein's field equation, the action-angle variables method was used to study the precession of planets, and the general M(?)ller transformation was discussed.Firstly, we gave an introduction of Riemann curvature tensor and the geodesic equation, which are very important for comprehending general relativity and solving Einstein's field equation. Secondly, canonical transformations, Hamilton-Jacobi theory and action-angle variables were investigated, which made a foundation to study the precession of planets. Thirdly, the fundamental principles and the basic mission of general relativity were introduced, and a brief introduction to a rising direction for general relativity - Numerical Relativity was presented.We concretely gave a detailed study of the perihelion precession of planets, in nearly-circular orbital and elliptical orbital respectively, with action-angle variables method. And, the explicit expressions of orbital precession were presented. After reviewed the Schwarzschild vacuum solution of Einstein's field equation, the motion equations of a particle in this field were investigated. Based on this set of self-contained equations, we presented the general integral form of action variables and discussed the properties of the angle variables in cases of nearly-circular orbital and elliptical orbital respectively. Then, the orbital precessions of the particle (a planet) in a cycle were come out by some analysis.On the other hand, we also gave a detailed discussion on the general M(?)ller transformation, which is a transformation from inertial reference frame to arbitrary time-varied reference system. And its differential form was em- phatically discussed. We considered a gravitational field in a reference system moving variably along a certain direction, the metric of which was derived later based on Einstein's equation. After the analysis and calculation, it came out that an inertial reference system can be determined by precisely checking the distantly separated watches, and the differential form of general M(?)ller transformation was presented.