Functional integral for systems with constraints that depend explicitly on the time

It is shown that the rules for constructing the functional integral in phase space for systems with singular Lagrangiaus proposed by Faddeev also remain valid when gauge conditions that depend explicitly on the time are used. Such conditions must be considered, for example, in the case when the canonical Hamiltonian in the theory is identically equal to zero (relativistic point particle, relativistic string, etc.). The functional integral is first expressed in terms of the physical canonical variables, for the separation of which a canonical transformation determined by the gauge conditions is used. In the case of nonstationary gauge conditions, the canonical transformation depends explicitly on the time. This leads to an additional (compared with the case considered by Faddeev) term in the Hamiltonian that determines the dynamics on the physical submanifold of the phase space.