Towards Energy Principles in the 18th and 19th Century – From D’Alembert to Gauss

Ekkehard Ramm, Universit¨at Stuttgart

The present contribution describes the evolution of extremum principles in mechanics in the 18th and the first half of the 19th century. First the development of the ’Principle of Least Action’ is recapitulated [1]: Maupertuis’ (1698-1759) hypothesis that for any change in nature there is a quantity for this change, denoted as ’action’, which is a minimum (1744/46); S. Koenig’s contribution in 1750 against Maupertuis, president of the Prussian Academy of Science, delivering a counter example that a maximum may occur as well and most importantly presenting a copy of a letter written by Leibniz already in 1707 which describes Maupertuis’ general principle but allowing for a minimum or maximum; Euler (1707-1783) heavily defended Maupertuis in this priority rights although he himself had discovered the principle before him. Next we refer to Jean Le Rond d’Alembert (1717-1783), member of the Paris Academy of Science since 1741. He described his principle of mechanics in his ’Trait´ede dynamique’ in 1743. It is remarkable that he was considered more a mathematician rather than a physicist; he himself ’believed mechanics to be based on metaphysical principles and not on experimental evidence’ [2]. Ne- vertheless D’Alembert’s Principle, expressing the dynamic equilibrium as the kinetic extension of the principle of virtual work, became in its Lagrangian ver- sion one of the most powerful contributions in mechanics. Briefly Hamilton’s Principle, denoted as ’Law of Varying Action’ by Sir William Rowan Hamilton (1805-1865), as the integral counterpart to d’Alembert’s differential equation is also mentioned. We then describe in detail the ’Principle of Least Constraint’ by Carl Friedrich Gauss (1777-1855), published in the Crelle Journal [3] which still exists today. The paper had only four pages with very little mathematics. Gauss defined the square of the differences between free and constrained accelerations for all masses as measure which should lead to a minimum constraint. Thus he rein- terpreted d’Alembert’s principle changing it into a minimum principle [4]. The principle has a direct analogy to his famous least square approximation. The publication was his only work on this subject. The Principles of Virtual Work (Displacements) and Complementary Work (Forces) became major building blocks in mechanics; they are related to the Principles of Minimum of Total Potential Energy (Green-Dirichlet) and Com- plementary Energy (Menabrea-Castilliano) once a potential exists. Furthermore mixed principles have been derived in the 20th century such as the Hellinger- Reissner or the Hu-Washizu Principle which became important starting points in Finite-Element Technology.

1 [1] I Szabo, Geschichte der mechanischen Prinzipien und ihrer wichtigsten Anwendungen, Birkh¨auser, Basel 1979

[2] JJ O’Connor and EF Robertson, Jean Le Rond D’Alembert in MacTutor History of Mathematics, Oct 1998 http://www-history.mcs.st-andrews.ac.uk/Biographies/D’Alembert.html

[3] CF Gauss, Uber¨ ein neues allgemeines Grundgesetz der Mechanik, Jour- nal fur¨ die reine und angew. Mathematik, herausg. v. CRELLE, Band IV (1829), 232–235

[4] C Lanczos, The Variational Principles of Mechanics, University of To- ronto Press, 3rd edition 1966, Section 8,106–110

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