Application of Invariant Variational Principles to the Optimal Design of a Column

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Abstract
Using Noether's technique we derive some invariants associated with the problems of optimization of design. While specifically a beam design problem is discussed here, the technique can be generalized in an obvious manner to thin plate, or shell problems. The specific problem discussed in this paper concerns the optimization of a column. Some invariants derived here as examples of the technique can be used to point out the basic difficulties discovered by much harder arguments in many recent papers. In particular, our method points out the existence of singularities which are associated with an optimal column design even in the absence of multiple eigenvalues in the Euler's eigenvalue problem, which corresponds to the buckling phenomenon.

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