Hamiltonian systems are a class of physical systems in classical mechanics that evolve according to Hamilton’s equations, which describe how the system’s coordinates and momenta change over time. These systems are defined by a special function called the Hamiltonian, usually representing the total energy (kinetic + potential) of the system.
Instead of using Newton’s laws (force-based), Hamiltonian mechanics provides an energy-based formulation that is especially powerful for complex and multi-body systems.
Hamilton’s Equations:
They relate time evolution to partial derivatives of the Hamiltonian:
- One equation gives how position changes over time.
- The other gives how momentum changes over time.
Key Features:
- Phase space formulation: Evolution occurs in a multi-dimensional space of positions and momenta.
- Conservation laws: Energy and other conserved quantities are naturally embedded in the system.
- Time-reversible: Solutions can be run forward or backward in time.
Examples:
- Planetary motion: The Hamiltonian describes gravitational interactions between celestial bodies.
- Harmonic oscillator: A mass on a spring has a quadratic Hamiltonian in position and momentum.
- Quantum mechanics: The Hamiltonian becomes an operator governing the evolution of quantum states.
Hamiltonian systems are essential in analytical mechanics, quantum mechanics, and chaos theory, offering a powerful and elegant framework for studying conservative (non-dissipative) physical systems.