Hamiltonian System

A Hamiltonian system is a dynamical system in which the total energy — the sum of kinetic and potential energies — is conserved over time, and the evolution of the system is governed by Hamilton's equations. Unlike dissipative systems, in which phase-space volume contracts and trajectories converge to attractors, Hamiltonian systems preserve phase-space volume exactly: the flow is incompressible, and information about initial conditions is never lost, only rearranged.

This conservation has profound consequences. Hamiltonian dynamics are time-reversible: if you reverse all velocities, the system retraces its path exactly. They possess no attractors — no fixed points, no limit cycles, no strange attractors — because volume preservation forbids the contraction that creates them. The solar system, ideal frictionless pendulums, and quantum mechanical wavefunctions (in their Schrödinger evolution) are all Hamiltonian.

The contrast with dissipative systems is not merely technical. It is metaphysical. Hamiltonian systems represent the classical ideal of a closed, deterministic universe in which the future is fully contained in the present. Dissipative systems represent the actual universe: open, irreversible, and generative. The dominance of Hamiltonian methods in theoretical physics for three centuries was not an accident. It was a methodological preference for problems that could be solved exactly over problems that described reality accurately.