👉 Hamiltonian mechanics is a reformulation of classical mechanics that uses the Hamiltonian function, a scalar quantity representing the total energy of a system, to describe its dynamics. Unlike Lagrangian mechanics, which focuses on the system's kinetic and potential energies and their derivatives, Hamiltonian mechanics centers on generalized coordinates and momenta, transforming the equations of motion into a set of first-order differential equations known as Hamilton's equations. These equations relate the time evolution of generalized coordinates and momenta to the system's energy, providing a powerful and elegant framework for analyzing complex mechanical systems. Hamiltonian mechanics simplifies many problems by exploiting symmetries and conservation laws, making it particularly useful in fields like quantum mechanics and statistical physics. It also offers a deeper geometric understanding of physical systems, linking them to phase spaces and symplectic geometry.
