Constraints from conservation laws in field theory

In this thesis, it has been shown, in three direct examples, that the Lagrangian formalism provides a successful approach to derive constraints on initial conditions from conservation laws. This achievement has been made possible thanks to the significant advancements in the theory of conservation laws for natural and gauge-natural field theories. The key factor enabling this success is the ability to split the Noether current $\mcl E$ into two parts, the reduced current $\wt \E$ and the superpotential $\U$. This decomposition ensures that conserved quantities are well-defined and that conservation laws on-shell take on a straightforward form, specifically, $\wt \E = 0$. The importance of these results lays in the fact that nor the Hamiltonian formalism, nor the canonical analysis have been used. While more work is needed to fully develop this approach, the examples presented in this thesis provide strong evidence that it has the potential to be a valuable tool for understanding the dynamics of gauge-natural theories.