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.
Highlights
Noether’s theorem, considered the most beautiful theorem in mathematical physics, attributes a conservation law to each symmetry [1]
We studied the Noether theorem for nonconservative field theories on time scales
After establishing Hamilton’s principle, we extracted from it the Lagrange equations on time scales
Summary
Noether’s theorem, considered the most beautiful theorem in mathematical physics, attributes a conservation law to each symmetry [1]. The studied systems are dissipative, regardless of whether they have applicability in physics or engineering This field of research is of particular interest due to the few existing contributions in the literature. In [16], the authors investigate the invariant properties of the discrete Lagrangian for conservative systems, the discrete analog of the calculus of variations and the Noether theorem, and another work [17] extends this theory to nonconservative systems. Noether conservation symmetries and laws for nonconservative discrete systems with irregular lattices were developed in [18], first by finding the discrete analog of Noether identities and by introducing the generalized quasi-external equations and their properties. In [19,20], the Noether symmetries and the conservation laws of nonconservative and nonholonomic mechanical systems on time scales were analyzed—a theory that unifies the two cases of continuous and discrete theories.
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