Corrected forms of physical supplementary conditions and gauge conditions in singular Lagrangian systems

It is shown that the corrected form of physical supplementary conditions in singular Lagrangian systems is that physical observables have weakly vanishing Poisson brackets with the elements of the minimum-evolution closed Poisson-bracket subalgebra of the first-class constraints, as well as with all the second-class constraints. A simple Cawley's example with the above features is studied. The origin of the gauge conditions is discussed and the corrected form and number of the gauge conditions in some more general singular Lagrangian systems are given. Our result differs from the accustomed conclusion which is that the number of gauge conditions or gauge freedoms is always equal to the number of first-class constraints. These results can provide a method for the quantization of some singular systems