The Calculus Of Variations Before The 19th Century

Based on intensive investigation of original sources and research literatures on the subject in consideration,and under the paradigm why mathematics was done in the history of mathematics,the dissertation studies systematically the origin and invention of the calculus of variations by means of source analysis and comparative argumentation.The combination with the Calculus,physics,especially variational principles and geometry is one of the features of tiffs dissertation.The main contributions are as follows:1.The prehistory of the calculus of variations is systematically examined by the analysis of the solutions of the classical isoperimetric problems and that of the development of the early least concept,and the limitations of the early geometric solutions,i.e.,the expression of the problems and the selection of comparative class, are pointed out.2.By investigating Newton's Problem of the Solid of Least Resistance,John Bernoulli's Brachystochrone Problem and Jacob Bernoulli's Isoperimetric Problem, the physics and Calculus background of the calculus of variations is illustrated,the fundamental thought together with the patterns of solutions are summarized,and the ideas and the interaction of the pioneers,such as the Bernoullis and B.Taylor,are studied.3.By the analysis of Euler's three earlier papers on the calculus of variations and that of his mistake on the fundamental principle,and the comparison with Euler's and his pioneers' solutions,the refinement of Euler's two core achievements,i.e., fundamental equation and isoperimetric rule,is explained.It is concluded that the Methodus inveniendi is not only a summary and improvement of earlier research,but an important breakthrough as to the early tradition.4.By elaborating Euler's general theory in the Methodus inveniendi,it suggests that the invariance of Euler's fundamental equation is a reflection of the change of the object of the Calculus.Meanwhile,two disputed questions are investigated and the limitations of Euler's method are discussed in order to illustrate the motive of Lagrange's research in this field.5.By investigating Euler's application of variational mathematics to mechanics, especially his principle of least action,the mechanics background of his research and its influence on Lagrange are revealed.6.By the discussion of Lagrange's innovation and generalization of the calculus of variations,the reason why Lagrange puts forward his method of variation by comparing Euler's method is explained.Besides,after the analysis of Lagrange's earlier contributions to the subject,the reasons that Lagrange changes hisδ-method from non-parametric form to parametric form are also examined.7.From the perspective of the variational discussion of Lagrange's contributions to the principles and analyzation of mechanics,the interaction between the calculus of variations and mechanics,especially the variational principles is made clear. 8.Based on Lagrange's algebraic program of the Calculus and his discussion on the integrable condition,his study on the foundations of the calculus of variations is investigated on one hand,and the mechanics origin of the multiplier in the calculus of variations is examined on the other.