Generalized Canonical Noether Theorem and Poincare-Cartan Integral Invariant

The constraints are invariant under the total variation of canonical variables including time, we can also deduce the classical canonical Noether theorem and Poincare-Cartan integral invariant for a system with a singular higher-order Lagrangian, which differs from the previous work to require that the constraints are invariant under the simultaneous variations of canonical variables. Based on the phase space generating function of Green function, the generalized first Noether theorem and Poincare-Cartan integral invariant in the quantum case for a system with a singular higher-order Lagrangian are derived. For the case in which the Jacobian of the transformation does not equal to unity, the quantal Poincare-Cartan integral invariant can be still derived. The comparisons of the results at the quantum level with those in classical theories are discussed.