The Australian Journal of Mathematical Analysis and Applications


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ISSN 1449-5910  

 

Paper Information

Paper Title:

A New Look at the Equations of the Calculus of Variations

Author(s):

Olivier de La Grandville

Faculty of Economics, Goethe University Frankfurt,
Theodore Adorno Platz 4, 60323 Frankfurt,
Germany.
E-mail: odelagrandville@gmail.com

Abstract:

We first offer an entirely new way to derive the celebrated Euler equation of the calculus of variations. The advantage of this approach is two-fold. On the one hand, it entirely eschews the two hurdles encountered by Lagrange, which become challenging in the case of elaborate functionals: getting rid of the arbitrary character of the perturbation given to the optimal function, and demonstrating the fundamental lemma of the calculus of variations. On the other hand, it leads in a direct way to the remarkable discovery made by Robert Dorfman ( 1969) when he introduced a modified Hamiltonian, which we called a Dorfmanian (2018) to honor his memory. In turn, extending the Dorfmanian enables to obtain readily the fundamental equations of the calculus of variations for the optimization of high-order functionals, or multiple integrals.

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