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J. F. Ganghoffer DIFFERENTIAL GEOMETRY, （j . f . Ganghoffer微分几何,）.pdf

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J. F. Ganghoffer DIFFERENTIAL GEOMETRY, （j . f . Ganghoffer微分几何,）

Rend. Sem. Mat. Univ. Pol. Torino - Vol. 65, 2 (2007) ISASUT Intensive Seminar J. F. Ganghoffer DIFFERENTIAL GEOMETRY, LEAST ACTION PRINCIPLES AND IRREVERSIBLE PROCESSES “Les phenomenes irreversibles et le theoreme de Clausius ne sont pas explicables au ´ ` ´ ´ ˆ ´ moyen des equations de Lagrange”. Poincare, 1908. ´ Abstract. This contribution is intended as both a course on differential geometry and an illustration of the involvement of differential geometry in the calculus of variations, in articu- lation with the occurrence of irreversibility. Potential applications in terms of the continuous symmetries of the constitutive laws of dissipative materials shall be mentionned, leading po- tentially to a systematic and predictive approach of the construction of the so-called master curves. 1. Introduction The contribution of differential forms to mathematics and physics is considerable, due to fact that they allow the uniﬁcation, generalization and conception of notions en- countered in a wide range of disciplines: mention amongst others elementary geom- etry, analysis, thermodynamics, continuum mechanics, electromagnetism, and analyt- ical mechanics, (see [1, 2, 9, 11, 12, 16, 26, 28, 29, 30, 31]). The ﬁrst part of the contribution gives the essentials of differential geometry in a synthetic manner. The proofs shall most of the time be omitted (the reader shall refer to one of the references related to differential geometry). The following notations shall be used in the sequel: the partial derivative of a quantity a with respect to the variable x shall be noted a , or a , or ∂ a . The transpose of a