On the complementary energy in elasticity and its history: the Italian school of nineteenth century

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In the formulation of mechanical theories, physicists usually neglect complementary energy, while rational mechanicians and engineers widely use it, especially in continuum physics and structural mechanics. Indeed, in many cases the solutions of elastic problems are found in a simpler way by resorting to complementary, rather than potential, energy. Moseley and Cotterill in England, Menabrea and Castigliano in Italy were among the first to introduce complementary energy in their papers, though implicitly; a more explicit formulation is in Crotti’s papers; and Engesser extended it to non-linear elasticity. In this work we run through the history of complementary energy and search for its possible mechanical meaning.