Generalized variational principle of Herglotz for several independent variables. First Noether-type theorem

Abstract

This paper extends the generalized variational principle of Herglotz to one with several independent variables and derives the corresponding generalized Euler–Lagrange equations. The extended principle contains the classical variational principle with several independent variables and the variational principle of Herglotz as special cases. A first Noether-type theorem is proven for the new variational principle, which gives the conserved quantities corresponding to symmetries of the associated functional. This theorem contains the classical first Noether theorem as a special case. As examples for applications we calculate a conserved quantity for the damped nonlinear Klein–Gordon equation and we show that the equations which describe the propagation of electromagnetic fields in a conductive medium can be derived from the generalized variational principle of Herglotz (but not from a classical variational principle).