Noether-Type Theorems and the Generalized Herglotz Principle in $q$-Contact Geometry

We develop a unified geometric framework for dissipative mechanical systems based on uniform $q$-contact manifolds, which provide an extended phase space equipped with multiple contact $1$-forms. Within this setting, we construct both Hamiltonian and Lagrangian formalisms and establish a generalized Noether-type theorem describing the relationship between symmetries and dissipated quantities. We further show that $q$-contact Lagrangian systems admit a genuine variational origin through a generalized Herglotz principle involving multiple action variables. The resulting $q$-contact Euler--Lagrange equations naturally depend on the scalar combination $\sum_{i=1}^q \partial L/\partial z_i$, reflecting the intrinsic structure of uniform $q$-contact geometry. We prove that this variational formulation is fully equivalent to the geometric $q$-contact Hamiltonian dynamics generated by the energy function. Several explicit examples involving multi-parameter dependent dynamics illustrate the effectiveness of the theory and demonstrate its potential to provide geometric insight into complex dissipative systems, thereby extending the scope of classical Lagrangian mechanics beyond symplectic and single-contact structures.