Abstract
We describe a necessary condition for the local solvability of the strong inverse variational problem in the context of Monge-Ampère partial differential equations and first-order Lagrangians. This condition is based on comparing effective differential forms on the first jet bundle. To illustrate and apply our approach, we study the linear Klein-Gordon equation, first and second heavenly equations of Plebański, Grant equation, and Husain equation, over a real four-dimensional manifold. Two approaches towards multisymplectic formulation of these equations are described.
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Notes
- 1.
By C∞(J1M)-linear we mean that the coefficients can be smooth functions and their first derivatives.
- 2.
Note that the minors of rank 1 recover all the second-order semi-linear differential equations, whilst the higher order minors (including the determinant of the whole matrix) add specific non-linear terms.
- 3.
Each ϕ ∈ C∞(M) defines a section \( M \to M \times \mathbb {R}\), x↦(x, ϕ(x)).
- 4.
We are using the summation convention of summing over the repeated indices.
- 5.
Cartan distribution exists also on higher jets but the first jets are special due to \(\operatorname {codim} \mathcal {C} = 1\).
- 6.
Note that (j1ϕ)∗k = k ∘ j1ϕ since k is a function.
- 7.
After the pullback by (j1ϕ)∗ and choice of the volume form on M.
- 8.
The equation can be reconstructed from the differential form via the M-A operator (8).
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Acknowledgements
This paper was written during my visit in Angers as a part of my PhD research, under the cotutelle agreement between the Masaryk University, Brno, Czech Republic, and the University of Angers, France. I am grateful for the funding provided by the Czech Ministry of Education, and by the Czech Science Foundation under the project GAČR EXPRO GX19-28628X, and I thank the University of Angers for the hospitality during the research period. I also want to express my gratitude to Volodya Rubtsov for his numerous valuable suggestions and detailed comments, and to Jan Slovák for clarification of concepts from the theory of jet bundles. The results were reported at the Winter School and Workshop Wisla 20-21, a European Mathematical Society event organized by the Baltic Institute of Mathematics.
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Suchánek, R. (2023). Some Remarks on Multisymplectic and Variational Nature of Monge-Ampère Equations in Dimension Four. In: Ulan, M., Hronek, S. (eds) Groups, Invariants, Integrals, and Mathematical Physics. Tutorials, Schools, and Workshops in the Mathematical Sciences . Birkhäuser, Cham. https://doi.org/10.1007/978-3-031-25666-0_3
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