Variational Theory: Variable-independence and Consistency

VARIATIONAL THEORY: VARIABLE-INDEPENDENCE AND CONSISTENCY Jian-zhong Zhao Geophysics Department, College of Resource, Environment and Earth Sciences, Yunnan University, Kunming, P.R. China 650091 [email protected] Abstract Variational theory of elasticity is surveyed in the context of mathematical logic in the present paper. The problem of variable-independence of variational principles raised by Chien is discussed. We find that Chien’s “High-order Lagrange Multiplier Theory”, which deals with the problem of variableindependence and constraint of variational principles, is inconsistent; Luo’s system, which is involved in the problem of variable-independence, is involved in contradictions; the conventional understanding of independence of variables of variational principles connotes contradiction. In the context of mathematical logic, variational theory must be established as a mathematical system of logic, excluding vagueness and misunderstanding. By consideration of logic, variableindependence is understood as identity of variables and then formalization of variational theory is a solution to the problem of variable-independence. Two consistent systems for elasticity, the Axiomatic System of Variation and the Formal System of Variation, are suggested in this paper.