Extension to velocity dependent gauge transformations

In previous papers it was shown that in phase space a generator of gauge transformations for a singular Lagrangian \(L(q,\dot q)\) is given byGε^α]=ε _k ^α φ _α ^k (q,p,t) where φ _α ^k are first class constraints andG is subjected to a stationarity condition. A non-trivial extension from velocity (or momentum) independent gauge transformations to velocity dependent ones is realized, by replacing gauge functions ε^α(t) with momentum dependent functions ε^α(q,p,t), as long asG satisfies the stationarity condition. Inversely, it is proved in a classical framework that, within velocity dependent gauge transformations, all generators of gauge transformations can be expressed in terms of the linear combinations of φ _α ^k .