A Hamiltonian system is a mathematical framework used in physics to describe the evolution of a dynamical system. It is based on the concept of a Hamiltonian, which is a function that represents the total energy of the system, including both kinetic and potential energy. The system's behavior is determined by Hamilton's equations, which provide a set of first-order differential equations governing the motion of the system. These systems are particularly important in classical mechanics and quantum mechanics, as they offer a powerful way to analyze complex systems. Hamiltonian systems are characterized by their conservation properties, meaning that certain quantities remain constant over time, making them useful for understanding the stability and long-term behavior of physical systems.