Deriving Variational Principles In Elasto-dynamics By Undetermined Lagrange Multiplier Method

Variational principle is a powerful tool in many scientific aspects. It has been widely used in mathematical physics, and especially it has been successfully used in solid mechanics. The Undetermined Lagrange Multiplier Method has been applied in variational calculus for more than 100 years, it has two modalities — — Undetermined Lagrange Multiplier Method joins variational operation and Undetermined Lagrange Multiplier Method doesn't join variational operation. The modality which Undetermined Lagrange Multiplier Method joins variationaloperation possesses two actions --releasing the restriction of variationals ofvariable functions confined by constraint conditions, and inverting the constrait conditions of the original variationals to stationary conditions of the new variationals; while the other modality only has the former action.By using the modality which Undetermined Lagrange Multiplier Method doesn't join variational operation, the stationary conditions of different forms variationals of Hamilton principle and complementary Hamilton principle in elasto-dynamics are derived; By using the other modality, generalized variational principles with two kinds of variables both with and without additional conditions are obtained, furthermore generalized variational principles with three kinds of variables are acquired.Then, a comparison is made between Undermined Lagrange Multiplier Method and Substitution Method, and it explains that these two methods have the same actions on cancelling and releasing restrictions.Finally, as examples, the motion equation of the thin board has been derived by Hamilton principle; And the dynamic displacement of the midspan of a simple beam which bears a dynamic vertical force on the center point has been calculated by finite-element method.