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Variational Theory: Variable-independence and Consistency

Variational theory of elasticity is surveyed in the context of mathematical logic in the present paper titled "Variational Theory: Variable-independence and Consistency" . The problem of variable-independence of variational principles raised by Professor Chien Wei-zang is discussed. We find that Chien’s “High-order Lagrange Multiplier Theory”, which deals with the problem of variable-independence 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, variable-independence 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.  Please see the details of the discussion in the attachment. 

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