phasespace

Phase space (also written phase-space) is the space of all possible states of a system, with each state represented by generalized positions and their conjugate momenta. For a system with n degrees of freedom, phase space has dimension 2n; a single point specifies the complete state at a given time, ignoring explicit time.

In classical mechanics, a particle in three dimensions has six phase-space coordinates (x, y, z, p_x, p_y, p_z). The system’s evolution is described by Hamilton’s equations, tracing a trajectory in phase space. Constraints reduce the effective dimension.

Liouville’s theorem states that the Hamiltonian flow preserves phase-space volume; the flow is a symplectomorphism. This leads to constant density of states along trajectories and underpins statistical mechanics.

Phase space is central to statistical mechanics, where integrals over positions and momenta define microcanonical and canonical ensembles, partition functions, and thermodynamic quantities. In many-body problems, phase-space methods underlie distribution functions and coarse-graining.

In quantum mechanics, phase space is not directly observable due to the uncertainty principle, but phase-space techniques exist. The Wigner distribution and Weyl transforms represent quantum states in phase space, providing a bridge between quantum and classical descriptions.

In dynamical systems and applied mathematics, phase space is used to study behavior via phase portraits, attractors, and Poincaré sections. The mathematical structure is often the cotangent bundle T*Q, equipped with a symplectic form, and canonical transformations preserve this structure.