Hamilton's equations

Hamilton's equations are a way to describe how a physical system evolves over time, especially in classical mechanics. They use two types of quantities: positions (where things are) and momenta (how fast they’re moving). These equations connect how the position and momentum change—how they evolve—based on a function called the Hamiltonian, which represents the total energy of the system. Essentially, Hamilton’s equations tell us how energy shifts between different forms and how the system progresses step-by-step, providing a powerful framework for analyzing motion in systems like planets, particles, or waves.