Hamiltonian mechanics

A formulation of classical mechanics in which generalized coordinates and conjugate momenta evolve according to Hamilton’s equations, usually generated by a Hamiltonian function.

Hamilton's equations: First-order differential equations that determine how generalized coordinates and conjugate momenta change over time. They express the motion generated by the Hamiltonian.
Lagrangian mechanics: A formulation of classical mechanics based on generalized coordinates and a Lagrangian function. The Legendre transform commonly converts its equations into Hamiltonian form.
William Rowan Hamilton: An Irish mathematician and physicist who developed Hamiltonian formulations of mechanics and optics. His work gave the formulation its name and foundational equations.
Celestial mechanics: The study of the motions of celestial bodies under gravity. Canonical variables and Hamiltonian perturbation methods analyze orbital motion.
Newtonian mechanics: Classical mechanics formulated through forces, masses, and second-order equations of motion. It describes the same classical trajectories without canonical phase-space variables.
Hamiltonian function: A function on phase space whose derivatives generate a system’s canonical equations of motion. Its derivatives specify the time evolution of coordinates and momenta.
Generalized coordinates: Independent variables that specify a mechanical system’s configuration without requiring Cartesian coordinates. They describe configurations before conjugate momenta are introduced.
Joseph-Louis Lagrange: An Italian-French mathematician and astronomer who formulated analytical mechanics using generalized coordinates and variational principles. Lagrange’s mechanics supplied the framework from which Hamiltonian mechanics developed.
Action-angle coordinates: Canonical coordinates in which integrable systems evolve with constant actions and linearly changing angles. They expose the simple long-term motion of integrable Hamiltonian systems.
Contact geometry: The geometry of odd-dimensional manifolds equipped with a maximally nonintegrable hyperplane distribution. It provides a natural geometric framework for dissipative systems, unlike ordinary Hamiltonian flow.

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