Singular transformations of coordinates

Author: saba beridze
Keywords: singular Lagrangian, Schwarzschild metric, geodesic
Annotation:

The thesis is devoted to the comparison of Hamilton-Jacobi and Euler-Lagrange methods in Schwarzschild space. The generalized Dirac method for constrained systems with singular Lagrangians, for which Legendre transformations are incorrect, is discussed. In the Schwarzschild metric, the Lagrangian of the geodesics is written and its singular nature is shown. We rewrite the Lagrangian of the geodesics in Eddington-Finkelstein coordinates, which are non-singular on the event horizon. It is shown that the Lagrangian singularity does not disappear in these coordinates, thus we confirm that the Euler-Lagrange and Hamilton-Jacob methods are not equivalent in the Schwarzschild metric. Then we proceed with Dirac's recipe, build a primary constraint and using Hamilton's equations we write equations of motion.