This thesis develops the theory of Lagrangian multiforms. For integrable equations which belong to compatible systems of equations, this theory encapsulates the entire system in a single variational principle.
We introduce definitions to distinguish between Lagrangian 2-forms and weak Lagrangian 2-forms in the discrete setting of quadrilateral stencils. We present three novel types of discrete Lagrangian 2-form for the integrable quad equations of the ABS list. Two of our new Lagrangian 2-forms have the quad equations, or a system equivalent to the quad equations, as their Euler-Lagrange equations, whereas the third produces the tetrahedron equations. This is in contrast to the well-established Lagrangian 2-form for these equations, which produces equations that are weaker than the quad equations (they are equivalent to two octahedron equations). We use relations between the Lagrangian 2-forms to prove that the system of quad equations is equivalent to the combined system of tetrahedron and octahedron equations.
We formulate the double zero property of Lagrangian multiforms in the discrete setting. For each of the discrete Lagrangian 2-forms associated with quad equations, we show that double zero expansions can be derived in terms of their respective corner equations.
We develop a framework to periodically reduce discrete Lagrangian 2-forms into weak discrete Lagrangian 1-forms. This framework elevates periodic reductions to the Lagrangian multiform level.
We link trigonometric functions to discrete and continuous Lagrangian 1-forms, making use of their addition formula. We derive discrete commuting flows for the McMillan equation (discrete autonomous Painlev´e II), derive Jacobi elliptic solutions and develop a discrete Lagrangian 1-form.
