Hamiltonian system

A Hamiltonian system is a dynamical system whose evolution is generated by a Hamiltonian function. Its equations relate changes in coordinates and momenta to derivatives of that function.

Hamilton’s equations: Differential equations that determine coordinate and momentum changes from derivatives of a Hamiltonian. They give the defining evolution law for a Hamiltonian system.
Generalized coordinates: Independent variables that specify a mechanical system’s configuration. Hamiltonian mechanics describes motion using these coordinates and their conjugate momenta.
Celestial mechanics: The study of the motions and gravitational interactions of celestial bodies. Orbital motion is often modeled with Hamiltonian equations and conserved quantities.
Dissipative system: A dynamical system in which phase-space volumes contract over time because of energy dissipation. Ordinary Hamiltonian flow preserves phase-space volume rather than contracting it.
Liouville’s theorem: The theorem that Hamiltonian flow preserves phase-space volume. It follows from the divergence-free character of Hamiltonian evolution.
Hamiltonian function: A function on a system’s state space that generates its time evolution. Its derivatives determine the system’s coordinate and momentum velocities.
Canonical momentum: The momentum variable conjugate to a coordinate in Hamiltonian mechanics. It pairs with each coordinate to specify a point in canonical phase space.
Pendulum: A mass suspended from a pivot that swings under gravity. Its idealized motion is a simple example of a Hamiltonian system.
Gradient flow: An evolution in which a state moves downhill along the gradient of a potential or energy. It typically decreases energy, unlike autonomous Hamiltonian motion.
Noether’s theorem: A theorem relating continuous symmetries of an action to conserved quantities. Symmetries in Hamiltonian mechanics yield conserved quantities through this connection.

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