E. Noether's variational symmetries and pluri-Lagrangian systems

Abstract: We analyze the relation of the notion of a pluri-Lagrangian system, which recently emerged in the theory of integrable systems, to the classical notion of variational symmetry, due to E. Noether. For finite-dimensional (classical mechanical) systems we show that, for any Lagrangian system with m commuting variational symmetries, one can construct a pluri-Lagrangian 1-form in the (m+1)-dimensional time, whose multi-time Euler-Lagrange equations coincide with the original system supplied with m commuting evolutionary flows corresponding to the variational symmetries. For two-dimensional hierarchies of Lagrangian PDEs, we show that if the flow of each PDE is a variational symmetry of all others, then there exists a pluri-Lagrangian 2-form for the hierarchy. The corresponding multi-time Euler–Lagrange equations coincide with the original system supplied with commuting evolutionary flows induced by the variational symmetries.

HEP - theorymathematical physics

Audience: researchers in the topic

Nagoya Math-Phys Seminar

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