The simplest variational problems (with free, fixed boundaries, the Bolz problem) in Banach spaces are considered. Necessary conditions for a local extremum in these problems are derived. An important class of Lagrangian mechanical systems is considered – local loaded fields, for which the Lagrangian has the form of an integral functional. Necessary conditions for the action functional – the Euler-Ostrogradsky equations and transversality conditions – are obtained. The equations of the theory of elasticity and Maxwell electrodynamics are derived from the variational principle for local fields.
lagrangian, action, local loaded field, variational problem, local extremum
Publication language:russian, pages:39
Mathematical modelling in actual problems of science and technics