Canonical coordinates and the canonical form of the two-form for a potential presentation: the coordinate expression $\omega ={\sum }_{a=1}^{m}d{\theta }^a\wedge d{\theta }^{m+a}$ with $m$ the chart's half-dimension, and the model's own momentum charts

Adopt the force model with potential presentation $(U,d)$ and mechanical Lagrangian $L=K-U$ of The action functional of a force model with a potential presentation: the mechanical Lagrangian, its chart instance and its dimensions with the momentum charts, the phase domains and the two-form datum of The canonical one-form and the two-form of a potential presentation: $\lambda =p\cdot dx$ on the Cartesian momentum phase chart and the induced forms on each generalized-coordinate chart: the Cartesian phase domain $W=Q\times {\mathbb{R}}^{3N}$ with $n=3N$, its two-form $\omega$, and for each generalized-coordinate map $X:\mathrm{\Omega }\to Q$ with $r$ coordinates the chart map ${\mathrm{\Phi }}_X$ and the form ${\omega }_X$ (Generalized-coordinate maps of the model, the substituted Lagrangian, the kinetic metric and generalized forces, The momentum of a potential presentation: the fibre derivative of the mechanical Lagrangian, the Cartesian momentum Mv and generalized momenta, Configuration domain of $N$ labeled particles: open admissible sets, regular holonomic constraints and the mass operator). All differentiability classes are those of $C^k$ Euclidean maps and diffeomorphisms.

(a) Canonical coordinates. Let $W_0\subseteq W$ (or $W_0\subseteq \mathrm{\Omega }\times {\mathbb{R}}^r$) be nonempty open, and let

$\theta =({\theta }^1,\dots ,{\theta }^{2m}):W_0⟶V\subseteq {\mathbb{R}}^{2m}$

be a $C^1$ chart of $W_0$ onto an open set $V$ (The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement), with $m$ the half-dimension of $W_0$. Write $\sigma =\omega {\mathrm{\mid }}_{W_0}$ and $\alpha =\lambda {\mathrm{\mid }}_{W_0}$ in the Cartesian case, and $\sigma ={\omega }_X{\mathrm{\mid }}_{W_0}$ and $\alpha ={\lambda }_X{\mathrm{\mid }}_{W_0}$ in the generalized-coordinate case. Each coordinate ${\theta }^a$ has a declared dimension ${\eta }_a$ and unit; its paired coordinate ${\theta }^{m+a}$ has dimension $\mu {\mathrm{\ell }}^2{\tau }^{-1}{\eta }_a^{-1}$ and a declared compatible unit, so every summand below has action dimension. Coordinate formulas use the numerical representatives in these fixed units. The coordinates $\theta$ are canonical coordinates for the presentation's two-form on $W_0$ when the two-form takes the canonical form

$\sigma ={\sum }_{a=1}^{m}d{\theta }^a\wedge d{\theta }^{m+a}$

pointwise on $W_0$, using the finite-coordinate form and pullback conventions of Finite-regularity differential forms, Hamiltonian fields and canonical maps in phase charts; equivalently, when the pullback identity ${\theta }^{\ast }{\omega }_V^{\mathrm{s}\mathrm{t}}=\sigma$ holds for the standard two-form ${\omega }_V^{\mathrm{s}\mathrm{t}}$ of $V$ (A phase domain, its canonical one-form and its standard symplectic two-form, Finite-regularity differential forms, Hamiltonian fields and canonical maps in phase charts), equivalently when $D\theta (z{)}^{\mathsf{T}}JD\theta (z)=J$ for the standard symplectic matrix $J$ at every $z\in W_0$ (A $C^1$ map between phase domains preserves the standard two-form iff $D{C}^{\mathsf{T}}JDC=J$, and a canonical diffeomorphism carries $X_{H\circ C}$ to $X_H$). A canonical chart of the presentation is a chart whose coordinates are canonical in this sense; the equivalence with the canonical maps defined later on this page is recorded with those definitions.

(b) The model's own charts are canonical. The Cartesian coordinates $(x,p)$ on $W$ are canonical coordinates: by clause 1 of The two-form of a potential presentation is closed and nondegenerate, and each generalized-coordinate momentum chart carries the same symplectic structure: ${\mathrm{\Phi }}_X^{\ast }\lambda =\sum P_{a}d{q}^a$ and ${\mathrm{\Phi }}_X^{\ast }\omega =\sum d{q}^a\wedge d{P}_a$ the two-form is $\omega ={\sum }_{a}d{x}^a\wedge d{p}_a$. For a generalized-coordinate map, the coordinates $(q,P)$ on $\mathrm{\Omega }\times {\mathbb{R}}^r$ are canonical coordinates for the pulled-back form ${\omega }_X$: by clause 2 of The two-form of a potential presentation is closed and nondegenerate, and each generalized-coordinate momentum chart carries the same symplectic structure: ${\mathrm{\Phi }}_X^{\ast }\lambda =\sum P_{a}d{q}^a$ and ${\mathrm{\Phi }}_X^{\ast }\omega =\sum d{q}^a\wedge d{P}_a$ one has ${\omega }_X={\sum }_{a}d{q}^a\wedge d{P}_a$, and when the map is a genuine chart ($r=3N$) the chart transition is a change of canonical coordinates on overlaps.

(c) Canonical one-form up to gauge at classical differentiability. For a $C^2$ canonical chart $\theta$, the source canonical one-form $\alpha$ specified in clause (a) from The canonical one-form and the two-form of a potential presentation: $\lambda =p\cdot dx$ on the Cartesian momentum phase chart and the induced forms on each generalized-coordinate chart is said to be in canonical form up to gauge in the canonical coordinates $\theta$ when the difference $\alpha -{\sum }_a{\theta }^{m+a}d{\theta }^a$ is a closed one-form on $W_0$; exactness of that difference is not required. For the presentation's own charts the difference of the two canonical one-forms is closed by clause 3 of The two-form of a potential presentation is closed and nondegenerate, and each generalized-coordinate momentum chart carries the same symplectic structure: ${\mathrm{\Phi }}_X^{\ast }\lambda =\sum P_{a}d{q}^a$ and ${\mathrm{\Phi }}_X^{\ast }\omega =\sum d{q}^a\wedge d{P}_a$. These charts are $C^2$ or smoother. For a merely $C^1$ chart clause (a) retains its full pointwise meaning, but the coefficients of the pulled-back one-form may be only continuous, so this clause does not apply the classical exterior derivative to them.

(d) Status and warnings. The item defines a coordinate condition on a declared chart and records the one fact that the presentation's own momentum charts satisfy it. It asserts no existence of canonical coordinates beyond the model's charts, no global statement on multiply connected phase domains, no dynamics and no measured value. A chart that is merely a diffeomorphism need not be canonical; the determinant test $\mathrm{\mid }\det D\theta \mathrm{\mid }=1$ is necessary but, in dimension $2m\ge 4$, not sufficient for canonicity (the scope remark of this page records the witness).