The intrinsic energy minimization in dynamical systems offers a valuable tool for minimizing the objective functions of computationally challenging problems in combinatorial optimization. However, most prior works have focused on mapping such dynamics to combinatorial optimization problems whose objective functions have quadratic degree (e.g., MaxCut); such problems can be represented and analyzed using graphs. However, the work on developing such models for problems that need objective functions with degree greater than two, and subsequently, entail the use of hypergraph data structures, is relatively sparse. In this work, we develop dynamical system-inspired computational models for several such problems. Specifically, we define the 'energy function' for hypergraph-based combinatorial problems ranging from Boolean SAT and its variants to integer factorization, and subsequently, define the resulting system dynamics. We also show that the design approach is applicable to optimization problems with quadratic degree, and use it develop a new dynamical system formulation for minimizing the Ising Hamiltonian. Our work not only expands on the scope of problems that can be directly mapped to, and solved using physics-inspired models, but also creates new opportunities to design high-performance accelerators for solving combinatorial optimization.