Adopted classical Hamiltonian dynamics and preparation

For a selected finite-dimensional nonrelativistic classical model, adopt a Hamiltonian phase datum Hamiltonian phase, complete-domain flow and reference measure and interpret each point of its manifold $M$ as a complete instantaneous model state. The Hamiltonian $H:M\to \mathbb{R}$ is the model's supplied primitive energy function; no derivation from a Lagrangian is assumed here. Choose a complete invariant domain $R$ and a prepared Borel probability measure ${\mu }_0$ on $R$. For each initial state $z_0\in R$, the unforced state curve is the smooth map $z_{z_0}:\mathbb{R}\to M$ defined by $z_{z_0}(t)={\mathrm{\Phi }}_t(z_0)$ using the flow from the phase datum; complete invariance ensures $z_{z_0}(t)\in R$ for every $t$. The preparation is a probability law over possible phase points; it is not an individual trajectory or a claim that an ensemble was experimentally sampled.

Use SI units for the physical interpretation. In a canonical cotangent chart, $q^i$ are generalized configuration coordinates and $p_i$ are the conjugate covector components, so ${\sum }_i{p}_{i}d{q}^i$ has units J s. If $q^i$ has unit $u_i$, then $p_i$ has unit J s/$u_i$. Thus Cartesian length coordinates have units m and their conjugates kg m/s, whereas dimensionless angular coordinates have conjugates in J s. Identifying a conjugate with mechanical momentum requires an additional model convention. The Hamiltonian has units J and time has units s. The reference action $a_0$ in the mathematical definition has units J s and is a normalization convention. If a canonical-temperature model is additionally selected, take $T>0$ as absolute temperature in K and adopt $k_B>0$ as a primitive energy-temperature conversion constant of this formulation, with units J/K. Set $\beta =(k_{B}T{)}^{-1}$ in J${}^{-1}$ and use the associated weight only when its partition integral is finite and positive. This postulate does not assert that the canonical law is normalizable for every Hamiltonian or that these adopted parameters describe nature.

This is an adopted model formulation and preparation rule. It is not derived from mathematical axioms, and citing statistical-mechanics treatments does not establish that a selected model describes nature. No ergodicity, equilibration, measured outcome, or stochastic bath is assumed.