Hello   
      
   The timelike geodesic in Schwarzschild spacetime can be written in GR:   
      
   E^2 -1 = (dr/d_tau)^2 -2GM/r +(L/r)^2 - (2GM L^2)/r^3    (1)   
      
   where E is the relativistic conserved energy of a unitary mass, on the   
   geodesic and L its conserved angular momentum.   
      
   In Newtonian mechanics the Hamiltonian is   
      
   e = 1/2 (dr.dt)^2 - GM/r + (1/2)(l/r)^2      (2)   
      
   where e is the total conserved energy on the geodesic of a unitary mass   
   and l its conserved angular momentum.   
      
   In weak field, for r >> rs, we may neglect the last term in   
   (1/r^3). Usually one divide equation 1 by 2 for getting a similar form.   
      
   (E^2 -1)/2 = (1/2)(dr/d_tau)^2 -GM/r + (1/2)(L/r)^2     (3)   
      
      
   But for convergence of equations (2) and (3), we have to assume that   
   e = (E^2 -1)/2.   
      
   That is the point that I do not understand: How an energy can be equal   
   to a square energy?   
      
   Thanks for help.   
   jacques   
      
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