This research introduces a class of non-parametric identifiers based on differential neural networks represented by Hamiltonian dynamics. The structure of the identifier corresponds to the form of a canonical Hamiltonian system that uses the evolution of generalized coordinates and momentums. The learning laws of the identifier come from applying the first integrals approach, which justifies the design of an exact identifier considering the time invariance of the Hamiltonian, with a finite number of activation functions in the identifier structure. The first integrals approach derives several learning laws for the proposed class of identifiers. The learning laws design uses the estimated derivative of the generalized momentum assessed via a super-twisting differentiator with multiple inputs and outputs. All proposed laws require the solution of differential continuous-time Riccati equations and nonlinear differential equations for the learning laws, which depend on the identification error and state constraints. The developed identifier was evaluated compared to an identifier that did not consider the Hamiltonian constraint using first integrals. This comparison included a numerical evaluation of the identifier considering its application to a classical Hamiltonian system associated with the Kepler dynamics representing satellite orbital evolution. This evaluation confirmed that the identification results were improved with the proposed learning laws regarding the class of Hamiltonian structures, and the quality indicators based on the mean square error were several times lower.