Second Noether-type theorem for the generalized variational principle of Herglotz

Abstract

The generalized variational principle of Herglotz defines the functional, whose extrema are sought, by a differential equation rather than by an integral. For such functionals the classical Noether theorems are not applicable. First and second Noether-type theorems which do apply to the generalized variational principle of Herglotz were formulated and proved. These theorems contain the classical first and second Noether theorems as special cases. We published the first Noether-type theorem previously in this journal. Here we prove the second Noether-type theorem and show that it reduces to the classical second Noether theorem when the Herglotz variational principle reduces to the classical variational principle.