Integrable Hamiltonian Hierarchies and Lagrangian 1-Forms

We present further developments on the Lagrangian 1-form description for one-dimensional integrable systems in both discrete and continuous levels. A key feature of integrability in this context called a closure relation will be derived from the local variation of the action on the space of independent variables. The generalised Euler-Lagrange equations and constraint equations are derived directly from the variation of the action on the space of dependent variables. This set of Lagrangian equations gives rise to a crucial property of integrable systems known as the multidimensional consistency. Alternatively, the closure relation can be obtained from generalised Stokes' theorem exhibiting a path independent property of the systems on the space of independent variables. The homotopy structure of paths suggests that the space of independent variables is simply connected. Furthermore, the N\"{o}ether charges, invariants in the context of Liouville integrability, can be obtained directly from the non-local variation of the action on the space of dependent variables.