Hamiltonian formulation

The Hamiltonian formulation is a way of describing a physical system's motion using energy. It focuses on two key quantities: position (where things are) and momentum (how fast and in what direction they’re moving). The Hamiltonian itself represents the total energy—both kinetic (motion) and potential (position-based). This approach transforms the equations of motion into a set of concise mathematical rules, making it easier to analyze complex systems in physics, from particles to planets. Essentially, it reframes how we understand and predict how things move by emphasizing energy and momentum.