CONSISTENCY OF VARIATIONAL THEORY

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 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 En’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 misunderstanding and uncertainty. 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.