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Phase space coordinates are canonical when their Poisson brackets have the form $\{q^I,p_J\}={\delta }_J^I$, with the position-position and momentum-momentum brackets zero.

The common zero locus of constraint functions $C_a$ in phase space is the constraint surface. A physical trajectory lies on this surface, but off-surface identities are often needed when analyzing a constraint algebra.

The direct product of integrable Hamiltonian systems is integrable. Pull each family's commuting integrals of motion back to the product phase space. Integrals from different factors have zero Poisson bracket because they use disjoint canonical coordinates, and their differentials are independent wherever the differentials within each factor are independent.

This spatially nonconstant local Maxwellian is constant along free characteristic curves. Its collision term vanishes by the collision invariants, so it solves the full Boltzmann equation. It is positive, rapidly decaying and integrable in phase space. Completing its velocity square gives local mean $\alpha tx/(\beta +\alpha t^2)$ and local variance proportional to $(\beta +\alpha t^2{)}^{-1}$, while its spatial profile broadens with time.

The Duhamel formula for Hamiltonian transport, phase space preservation and Minkowski integral inequality give the displayed bound. A time-independent source in an Lp space yields finite space-time norm on every bounded interval; it need not yield integrability on infinite time.

A galactic distribution function gives stellar mass or number per phase space volume. Integrating over velocity gives spatial density, while velocity moments and projection along the line of sight predict kinematic observables.

If $H$ is $C^2$ and its spatial Hessian matrix is uniformly bounded on every finite time interval, its Hamiltonian vector field is globally Lipschitz continuous in phase space with at most linear growth. The Picard-Lindelof theorem and Gronwall inequality give a unique Hamiltonian flow for all finite forward and backward times. Bounds on ${\nabla }_{z}H(t,0)$ follow from time continuity on compact intervals.

Along Hamilton's equations, spatial derivatives of the Hamiltonian function cancel in its total time derivative. Thus $dH/dt={\partial }_{t}H$. An autonomous Hamiltonian function is conserved; explicit time dependence generally prevents conservation even though phase space is still preserved.

An integral of motion is a function on phase space that remains constant along every trajectory. If it has no explicit time dependence, $I(q,p)$ is an integral exactly when its Poisson bracket with the Hamiltonian vanishes: $\{I,H\}=0$.

Describe many-particle systems through distributions on phase space. Transport moves particles along trajectories, while collision or diffusion terms modify their velocity distributions. The Boltzmann equation, Hamiltonian Liouville equation and Fokker-Planck equation are central models.

The mass-normalized first velocity moment of a distribution on phase space. For a velocity-only distribution, omit the position integral. Nonzero finite mass and a finite first moment are required. For a probability distribution this is its expected value; its unnormalized counterpart is the momentum.