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arXiv · 2412.14700

On the geometry of Lagrangian one-forms

Vincent Caudrelier·Derek Harland
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Abstract

Lagrangian multiform theory is a variational framework for integrable systems. In this article we introduce a new formulation which is based on symplectic geometry and which treats position, momentum and time coordinates of a finite-dimensional integrable hierarchy on an equal footing. This formulation allows a streamlined one-step derivation of both the multi-time Euler-Lagrange equations and the closure relation (encoding integrability). We argue that any Lagrangian one-form for a finite-dimensional system can be recast in our new framework. This framework easily extends to non-commuting flows and we show that the equations characterising (infinitesimal) Hamiltonian Lie group actions are variational in character. We reinterpret these equations as a system of compatible non autonomous Hamiltonian equations.

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Vincent Caudrelier, Derek Harland. 2024-12-19. On the geometry of Lagrangian one-forms. https://doi.org/10.1007/s11005-025-01925-0

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