Higher-order and non-geometric symmetries and Noether's theorem in k-cosymplectic Hamiltonian field theory

In this paper, we investigate higher-order symmetries and non-geometric symmetries within the framework of $k$-cosymplectic Hamiltonian field theory. We introduce the concept of generalized infinitesimal symmetry, which relaxes the standard symmetry condition, and then define two new classes of symmetries: higher-order general infinitesimal Cartan symmetries and good non-geometric symmetries. For each of these symmetry classes, we establish a Noether-type theorem that provides explicit conserved quantities for exact $k$-cosymplectic Hamiltonian systems. These results extend and generalize previous work on standard Cartan symmetries in the $k$-cosymplectic setting. Furthermore, we extend all of these results in a parallel fashion to Hamiltonian systems on $k$-symplectic manifolds, demonstrating the unifying nature of our approach. Concrete examples are provided to illustrate the existence and applicability of the proposed symmetries. Our framework offers a systematic geometric treatment of generalized symmetries and their associated conservation laws in classical field theories.