Some variational formulations for continuum nonlinear dynamics
The classic approach to the establishment of a variational principle requires a selfadjoint system. Here the authors give an example that even a simple equation of a motion without friction with prescribed initial conditions is not selfadjoint with respect to classical bilinear forms, leading to difficulties in searching for classical variational principles for dynamical problems. Hamilton principle can deal with dynamical problems only for conditions prescribed at the beginning and at the end of the motion, so the Hamilton principle cannot be used for initial conditions. The authors introduce an auxiliary system and establish a classical variational principle with original fields and auxiliary fields, converting initial conditions into natural ones, which can make the obtained variational principle stationary. Three variational principles with auxiliary variables are established for dynamical problems of elasticity. However, the disadvantage of doubling of the unknown variables in the functional leads to inconvenience when incorporating numerical simulations.
scientific article; zbMATH DE number 1441406