After a brief review of Lagrangian multiform theory as a variational framework to describe integrable systems, I will focus on Lagrangian one-forms which deal with finite dimensional integrable systems. Introducing a gauging procedure within

this framework allows for a variational description of the well-established Hamiltonian reduction technique to produce integrable hierarchies. This enables us to adapt Hitchin's original construction (which I will briefly review) to the Lagrangian setting and to obtain for the first time the variational description of the Hitchin system. It turns out that doing so unravels a direct connection with holomorphic-topological gauge theory which has triggered a lot of attention as a gauge theoretic framework to construct integrable theories. Specifically, in our case, we construct a gauged Lagrangian one-form for the so-called 3d mixed BF theory with defects. Special cases of our construction yield Lagrangian one-forms for (perhaps) more familiar integrable systems such as (elliptic) Gaudin and spin Calogero models. This is based on joint work with D. Harland, A.A. Singh and B. Vicedo. 

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