phase space

Phase space is the arena in which a mechanical system's entire state is a single point. Ordinary space tells you where the parts are; configuration space collects all the coordinates into one point; but neither tells you what happens next, because you also need to know how fast everything is moving. Phase space fixes this by giving every degree of freedom two axes -- a position and its conjugate momentum -- so that one point encodes the complete instantaneous state and the future is thereafter determined.

For a system with n degrees of freedom, phase space is the 2n-dimensional space with coordinates (q_1, ..., q_n, p_1, ..., p_n). A single point is a full initial condition; Hamilton's equations turn that point into a unique trajectory, so exactly one path passes through each point and trajectories never cross. The collection of all trajectories is the phase portrait of the system. Phase space is not just configuration space plus velocities: the momenta p are the natural (canonically conjugate) variables, they transform differently from velocities, and the geometry that lives here -- the symplectic structure dq ^ dp -- is what makes the dynamics canonical.

This is the setting where mechanics connects to statistics and to the quantum world. Statistical mechanics replaces one point by a probability cloud (an ensemble) drifting through phase space, and Liouville's theorem says the cloud moves like an incompressible fluid. Quantum mechanics coarse-grains phase space into cells of size roughly h per degree of freedom, which is why h^n appears in counting states. Thinking in phase space, rather than in position alone, is the single biggest conceptual shift of the Hamiltonian formulation.