### 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.

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.

#### Keywords

Noether's theorems; variational principles; conserved quantities

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