*Article* **Symmetries for Nonconservative Field Theories on Time Scale**

**Octavian Postavaru \*,† and Antonela Toma †**

Center for Research and Training in Innovative Techniques of Applied Mathematics in Engineering,

University Politehnica of Bucharest, 060042 Bucharest, Romania; antonela.toma@upb.ro

**\*** Correspondence: opostavaru@linuxmail.org; Tel.: +40-0770-241-912

† These authors contributed equally to this work.

**Abstract:** Symmetries and their associated conserved quantities are of great importance in the study of dynamic systems. In this paper, we describe nonconservative field theories on time scales—a model that brings together, in a single theory, discrete and continuous cases. After defining Hamilton's principle for nonconservative field theories on time scales, we obtain the associated Lagrange equations. Next, based on the Hamilton's action invariance for nonconservative field theories on time scales under the action of some infinitesimal transformations, we establish symmetric and quasi-symmetric Noether transformations, as well as generalized quasi-symmetric Noether transformations. Once the Noether symmetry selection criteria are defined, the conserved quantities for the nonconservative field theories on time scales are identified. We conclude with two examples to illustrate the applicability of the theory.

**Keywords:** time scales; Noether theory; conserved quantity
