Knowra Phase space Phase space A space whose points represent possible states of a system, with each point specifying enough variables to determine its evolution. A system’s motion is represented by a trajectory through this space.
State space : A set of all possible states of a system, represented using variables that distinguish one state from another. Phase space is a state space whose coordinates commonly include positions and momenta.
Hamiltonian mechanics : A formulation of classical mechanics that describes evolution using coordinates, conjugate momenta, and a Hamiltonian. Its canonical equations specify the flow of points through mechanical phase space.
Statistical mechanics : A framework connecting microscopic states and dynamics with macroscopic thermodynamic behavior. It uses distributions over phase space to calculate equilibrium properties.
Configuration space : A space whose points represent the possible configurations of a system, without specifying its momenta. Unlike phase space, it omits momentum variables and therefore does not alone specify general mechanical evolution.
Generalized coordinates : Independent variables that specify a mechanical system’s configuration, such as angles or positions. They provide coordinates for describing configurations within mechanical phase space.
Hamiltonian : A function that generates time evolution in Hamiltonian mechanics, often equal to total energy. Its derivatives determine how phase-space coordinates and momenta change with time.
Microcanonical ensemble : A statistical ensemble of isolated systems with fixed energy, particle number, and volume. Its states occupy an energy surface within phase space.
Energy surface : The subset of phase space on which a system’s conserved energy has a fixed value. For a fixed-energy system, motion is confined to this lower-dimensional part of phase space.
Generalized momentum : A momentum variable conjugate to a generalized coordinate in Lagrangian and Hamiltonian mechanics. Pairing each coordinate with its conjugate momentum gives Hamiltonian phase-space coordinates.
Hamilton's equations : First-order differential equations governing canonical coordinates and momenta in Hamiltonian mechanics. They define the local direction of phase-space trajectories.
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