Hamiltonian Mechanics: The Energy View of Motion

Hamiltonian mechanics reformulates physics by focusing on energy rather than forces.
The Hamiltonian represents the total energy of a system—kinetic plus potential.
It describes motion using positions and momenta instead of accelerations.
Hamilton’s equations provide a powerful alternative to Newton’s laws.
Phase space plots position and momentum, revealing the full state of a system.
Planetary orbits, oscillators, and quantum particles can all be modeled with Hamiltonians.
In quantum mechanics, the Hamiltonian operator determines how systems evolve over time.
Symmetries in the Hamiltonian link directly to conservation laws.
Hamiltonian mechanics provides the foundation for statistical mechanics and thermodynamics.
At its core, it shows motion as the natural unfolding of energy in time.