Can one derive the Lagrange eqn based on the methods of Lagrange multipliers? That is, we need to minimize the action with respect to the trajectory keeping the net energy of the body in motion constant. The conservation of energy will impose suitable constraint. Now if we consider the action to be function of position and momenta the equating the differential of the Lagrangian (in sense of minimizing) with respect to the same to zero, can we obtain a suitable expression for action?
Comments to the post (v2):
For a system with only holonomic (and semi-holonomic) constraints it is possible to derive Lagrange equations from Newton's laws, cf. e.g. Herbert Goldstein, Classical Mechanics, chapter 1 & 2; this & this Phys.SE posts.
Total energy conservation alone only directly implies EOM for a system with 1 DOF.