Generalized canonical Noether theorem and Poincare-Cartan integral invariant for a system with a singular high-order Lagrangian and an application

Based on the canonical action, a generalized canonical first Noether theore m and Poicare-Cartan integral-invariant for a system with a singular high-o rder Lagrangian are derived. It is worth while to point out that the constr aints are invariant under the total variation of canonical variables includ ing time. We can also deduce the result which differs from the previous wor k to require that the constraints are invariant under the simultaneous vari ations of canonical variables. A counter example to a conjecture of the Dir ac for a system with a singular high-order Lagrangian is given, in which th ere is no linearization of constraint.