Hamiltonian dynamics is a branch of classical mechanics that deals with the study of the motion of systems with a large number of degrees of freedom. It is based on the Hamiltonian formalism, which uses a function called the Hamiltonian to describe the system's dynamics. The Hamiltonian is defined as the sum of the system's kinetic and potential energies. In Hamiltonian dynamics, the equations of motion are derived from Hamilton's equations, which describe how the system's state evolves over time. These equations are usually written in terms of canonical coordinates and momenta, which are related to the system's position and velocity. The Hamiltonian formalism is particularly useful for studying systems in which conservation of energy plays a significant role, such as celestial mechanics and quantum mechanics. It also provides a powerful framework for analyzing the stability and behavior of dynamical systems. Overall, Hamiltonian dynamics provides a systematic and elegant approach to understanding the behavior of complex systems in classical mechanics.