Lagrangian explicitly preserves symmetries of a theory?

[copernicus1]

How are the Hamiltonian and Lagrangian different as far as preserving symmetries of a theory? Peskin and Schroeder write that the path integral formalism is nice because since it's based on the action and Lagrangian it explicitly preserves all the symmetries, but I'm wondering how/why the Hamiltonian doesn't. I know H isn't invariant under Lorentz transformations, but isn't it true that quantities commuting with H are conserved? So can't you find other symmetries of the system this way using the Hamiltonian? Or are they mainly referring to the Lorentz invariance of the Lagrangian (density) and action?

Thanks!

 

[jfy4]

The two formalisms are equivalent. However, the hamiltonian formulation singles out the time variable through the definition of the canonical momentum. Any time you single something out you break a symmetry. However, while explicit lorentz invariance is broken by this, it can be recovered.