Lagrangian and Hamiltonian Mechanics

Lagrangian and Hamiltonian mechanics are reformulations of Newtonian mechanics that focus on energy rather than forces. The Lagrangian method, developed by Joseph-Louis Lagrange, uses a quantity called the Lagrangian (kinetic energy minus potential energy) to derive the equations of motion. Hamiltonian mechanics, created by William Rowan Hamilton, builds on this by using the Hamiltonian (total energy) and introduces a powerful mathematical framework involving coordinates and momenta. Both approaches offer insights into complex systems and are widely used in advanced physics, particularly in fields like quantum mechanics, where they simplify calculations and provide deeper understanding.