Hamiltonian mechanics is a way to describe how physical systems, like planets or pendulums, move over time, focusing on energy rather than just forces. By reframing complex dynamics through energy lenses, this 19th-century physics framework now powers cutting-edge generative AI. It uses generalized coordinates \( q \) (like position) and their conjugate momenta \( p \) (related to momentum), forming a phase space that captures the system's state. This approach is particularly useful for complex systems with many parts, making it easier to find patterns and conservation laws.