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Conservative chaotic systems (CCSs) have unique advantages over dissipative chaotic systems (DCSs) in the fields of secure communication and pseudo-random number generators (PRNGs), etc. However, there are relatively fewer reports on CCSs than DCSs. To this end, this paper proposes an effective method for constructing a family of Hamiltonian conservative chaotic systems (HCCSs) by letting any three of the four sub-bodies denoted by 4D generalized Euler equations share a rotation axis. From theoretical analysis to experimental verification, one of the proposed HCCSs is studied thoroughly to demonstrate the effectiveness of this method. The example system has an infinite number of equilibrium points, which belong to either centers or saddles, resulting in hidden chaos. Besides, through the bifurcation diagram, parametric chaotic set, and Lyapunov exponent, richly dynamic behaviors related to parameters are displayed. Moreover, the 3D phase portraits verify that the Hamiltonian energy is conservative. Numerous energy-related coexisting orbits are discovered in this system, such as the coexistence of quasi-periodic orbits, chaotic orbits, and chaotic quasi-periodic orbits. Furthermore, the breadboard-based circuit is implemented to illustrate the HCCS’s physical feasibility. Finally, the PRNG based on the HCCS has excellent randomness in terms of NIST and TESTU01 test results. [ABSTRACT FROM AUTHOR]