Integrable Lagrangians and modular forms
Abstract
We investigate non-degenerate Lagrangians of the form ∫f(ux,uy,ut) dx dy dt such that the corresponding Euler-Lagrange equations ()x+()y+()t=0 are integrable by the method of hydrodynamic reductions. We demonstrate that the integrability conditions, which constitute an involutive over-determined system of fourth order PDEs for the Lagrangian density f, are invariant under a 20-parameter group of Lie-point symmetries whose action on the moduli space of integrable Lagrangians has an open orbit. The density of the 'master-Lagrangian' corresponding to this orbit is shown to be a modular form in three variables defined on a complex hyperbolic ball. We demonstrate how the knowledge of the symmetry group allows one to linearize the integrability conditions.
- Publication:
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Journal of Geometry and Physics
- Pub Date:
- June 2010
- DOI:
- 10.1016/j.geomphys.2010.02.006
- Bibcode:
- 2010JGP....60..896F