a Hamiltonian system
/ ham-il-TOH-nee-an /
Imagine a frictionless world where energy is never lost: a pendulum in a vacuum swings forever, a planet orbits without ever spiralling in. In such a world the total energy is fixed, and the motion is permanently trapped on a surface of constant energy. A Hamiltonian system is the precise framework for this kind of energy-conserving motion, organized around a single function — the Hamiltonian — that plays the role of total energy.
Formally, the state splits into a position x and a momentum y, and the equations take a special paired form: x' = ∂H/∂y, y' = -∂H/∂x, where H(x,y) is the Hamiltonian. The clever pairing forces energy to be conserved: along any trajectory, H' = (∂H/∂x) x' + (∂H/∂y) y' = (∂H/∂x)(∂H/∂y) + (∂H/∂y)(-∂H/∂x) = 0. So H stays exactly constant on every solution — the trajectories are precisely the level curves of H, the contour lines of the energy landscape. The flow also preserves area in the phase plane (Liouville's theorem), so it can neither contract toward a sink nor expand from a source.
Because energy cannot decay, a Hamiltonian system has no asymptotically stable equilibria and no limit cycles — every nondegenerate equilibrium is either a saddle (an energy pass) or a center (an energy minimum or maximum surrounded by closed orbits). This is the deep contrast with gradient systems, where energy only decreases and motion always descends. Hamiltonian mechanics is the backbone of physics — celestial mechanics, optics, and the bridge to quantum theory — and explains why an undamped nonlinear pendulum has a genuine center (its energy contours really do close up) rather than the spiral that any small damping would create.
The undamped pendulum x' = y, y' = -sin(x) is Hamiltonian with H = (1/2)y^2 + (1 - cos x); trajectories follow constant-H curves — closed loops (oscillation) for low energy, wavy lines (full rotation) for high energy.
Energy H is conserved, so trajectories are its level curves; equilibria are centers or saddles, never sinks.
Conservation of energy is exactly why a Hamiltonian system can have NO asymptotically stable equilibrium and no limit cycle — energy would have to decrease for that. Add the slightest damping and the system stops being Hamiltonian, turning centers into spirals.