Hamilton's equations are a set of first-order differential equations used in classical mechanics to describe the evolution of a physical system. They are derived from the Hamiltonian, which represents the total energy of the system, and provide a powerful framework for analyzing motion and dynamics. In Hamilton's formalism, the state of a system is described by generalized coordinates and momenta. The equations consist of two main parts: one for the time evolution of the coordinates and another for the momenta. This approach is particularly useful in fields like quantum mechanics and statistical mechanics, where it simplifies complex problems.