Constraints and gauge relations in the Lagrangian formalism

For several Lagrangians we show that their local symmetries can be obtained from the associated Euler-Lagrange equations, and we exhibit the explicit presence of the genuine constraints into gauge identities. We also employ the Lanczos approach to Noether’s theorem to give connections between the genuine constraints and their time derivatives. Besides, it is evident that the Hamiltonian secondary and tertiary constraints have relationship with the genuine constraints.