ABSTRACT
The big advantage of using an action principle is that S is a Lorentz scalar, which makes it much easier to guarantee Lorentz covariance. As the action will be the starting point of the path integral formulation of field theory, Lorentz covariance is much easier to establish within this framework. It is well known that the Hamiltonian equations imply that H itself is conserved with time, provided the Lagrangian or Hamiltonian has no explicit time dependence. Conservation of energy is one of the most important laws of nature, and it is instructive to derive it more directly from the fact that L does not depend explicitly on time. Both conservation of momentum and energy are examples of conservation laws that are consequence of symmetries, translation and time invariance.
