This is Mats Vermeeren's talk "The multisymplectic form formula for Lagrangian PDEs" at the BMS Student Conference 2014. The talk was given on Thursday, February 20, 2014, 10:00am at Technische Universität Berlin.
The symplectic two-form lies at the heart of classical mechanics. It is intrinsic in the Hamiltonian formulation of mechanics, and is very relevant in the numerical analysis of mechanical systems. It can also be derived in a cute way from the Lagrangian formulation of mechanics.
An immediate consequence of this derivation is that the symplectic two-form is conserved by the flow. This approach generalizes to field theories, where one deals with Lagrangian PDEs instead of Lagrangian ODEs. The conservation law that results from this generalization is called the multisymplectic form formula. Although it has been known in the literature for about 15 years, it is still unclear what the optimal way to formulate it is.
I will begin this talk with a quick review of the Euler-Lagrange equations and the symplectic property in mechanics. Then I will move on to field theories. I will state the multisymplectic form formula as it can be found in the literature and explain its shortcomings. Finally, I will discuss how the concept of the variational bicomplex provides the tools to write the multisymplectic form formula in a more satisfactory form.
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