# A symplectic Brezis-Ekeland-Nayroles principle

We submitted to the arXiv the article

Marius Buliga, Gery de Saxce, A symplectic Brezis-Ekeland-Nayroles principle

You can find here the slides of two talks given in Lille and Paris a while ago,  where the article has been announced.

UPDATE: The article appeared, as  arXiv:1408.3102

This is, we hope, an important article! Here is why.

The Brezis-Ekeland-Nayroles principle appeared in two articles from 1976, the first by Brezis-Ekeland, the second by Nayroles. These articles appeared too early, compared to the computation power of the time!

We call the principle by the initials of the names of the inventors: the BEN principle.

The BEN principle asserts that the curve of evolution of a elasto-plastic body minimizes a certain functional, among all possible evolution curves which are compatible with the initial and boundary conditions.

This opens the possibility to find, at once the evolution curve, instead of constructing it incrementally with respect to time.

In 1976 this was SF for the computers of the moment. Now it’s the right time!

Pay attention to the fact that a continuous mechanics system has states belonging to an infinite dimensional space (i.e. has an infinite number of degrees of freedom), therefore we almost never hope to find, nor need the exact solution of the evolution problem. We are happy for all practical purposes with approximate solutions.

We are not after the exact evolution curve, instead we are looking for an approximate evolution curve which has the right quantitative approximate properties, and all the right qualitative exact properties.

In elasto-plasticity (a hugely important class of materials for engineering applications) the evolution equations are moreover not smooth. Differential calculus is conveniently and beautifully replaced by convex analysis.

Another aspect is that elasto-plastic materials are dissipative, therefore there is no obvious hope to treat them with the tools of hamiltonian mechanics.

Our symplectic BEN principle does this: one principle covers the dynamical, dissipative evolution of a body, in a way which can be reasonably easy amenable to numerical applications.

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