A phase-space Lagrangian one-form with a one-step variational principle recovers the multi-time Euler-Lagrange equations and closure, and its Lie-group version makes Hamiltonian group actions variational.
Suris, Variational formulation of commuting Hamiltonian flows: mu lti-time Lagrangian 1-forms, J
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On the geometry of Lagrangian one-forms
A phase-space Lagrangian one-form with a one-step variational principle recovers the multi-time Euler-Lagrange equations and closure, and its Lie-group version makes Hamiltonian group actions variational.