Hamilton's Equations are a set of first-order differential equations that describe the evolution of a physical system in classical mechanics. They are derived from the Hamiltonian, which represents the total energy of the system, combining both kinetic and potential energy. These equations provide a powerful framework for analyzing dynamic systems, especially in areas like mechanics and optics. In Hamilton's formalism, the state of a system is defined 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 theoretical physics and has applications in quantum mechanics and chaos theory.