  We describe a variational framework for non-commuting flows, extending the
theories of Lagrangian multiforms and pluri-Lagrangian systems, which have
gained prominence in recent years as a variational description of integrable
systems in the sense of multidimensional consistency. In the context of
non-commuting flows, the manifold of independent variables, often called
multi-time, is a Lie group whose bracket structure corresponds to the
commutation relations between the vector fields generating the flows. Natural
examples are provided by superintegrable systems for the case of Lagrangian
1-form structures, and integrable hierarchies on loop groups in the case of
Lagrangian 2-forms. As particular examples we discuss the Kepler problem, the
rational Calogero-Moser system, and a generalisation of the
Ablowitz-Kaup-Newell-Segur system with non-commuting flows. We view this
endeavour as a first step towards a purely variational approach to Lie group
actions on manifolds.
