The structure of time and inertial forces in Lagrangian mechanics
  Classically time is kept fixed for infinitesimal variations in problems in mechanics. Apparently, there appears to be no mathematical justification in the literature for this standard procedure. This can be explained canonically by unveiling the intrinsic mathematical structure of time in Lagrangian mechanics. Moreover, this structure also offers a general method to deal with inertial forces. 
J. Mu\~noz D\'iaz