  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.
