Canonical maps of a potential presentation: pullback-preserving immersions and the equal-dimensional determinant and field-transport tests

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 and 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 canonical coordinates of 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 and the Hamiltonian fields of The Hamiltonian field of a phase function for a potential presentation: the unique field $X_f$ with ${\iota }_{X_f}\omega =df$, and $X_H=(H_p,-H_x)$ for the branch Hamiltonian (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). Let $W_j\subseteq {\mathbb{R}}^{2m_j}$ be nonempty open subsets of phase domains of the presentation, carrying their respective two-forms ${\omega }_j$ 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 (for a generalized-coordinate domain this is the pulled-back form ${\omega }_X$ in its own $(q,P)$ coordinates). The dimensions $2m_1$ and $2m_2$ need not agree.

(a) Pullback-preserving maps and transformations. A $C^1$ map $C:W_1\to W_2$ is a canonical map of the presentation when

$C^{\ast }{\omega }_2={\omega }_1\text{on }{W}_1$

with pointwise pullback as in The pullback of a differential form. In standard phase coordinates this is the dimension-aware matrix identity

$DC(z{)}^{\mathsf{T}}{J}_{m_2}DC(z)=J_{m_1},$

where $DC(z)$ is a $2m_2\times 2m_1$ matrix and $J_m=\left(\begin{array}{cc}0& I_m\\ -I_m& 0\end{array}\right)$. It forces $DC(z)$ to be injective: if $DC(z)v=0$, then $v^{\mathsf{T}}{J}_{m_1}w=0$ for every $w$, so $v=0$ by nondegeneracy of $J_{m_1}$, so $m_1\le m_2$. Thus it is a $C^1$ immersion preserving the symplectic forms (a symplectic immersion). If it is also a $C^1$ embedding of the underlying manifolds (injective and a homeomorphism onto its image), call it a canonical embedding; pullback preservation alone asserts no global injectivity or topological embedding.

When $m_1=m_2$, a canonical map is automatically a local diffeomorphism by the inverse function theorem; call it a local canonical transformation. A canonical transformation is a canonical map that is a $C^1$ diffeomorphism onto $W_2$. The same-dimensional definitions agree with the finite-regularity convention of Finite-regularity differential forms, Hamiltonian fields and canonical maps in phase charts; the $C^{\mathrm{\infty }}$ case agrees with Canonical transformation and Symplectomorphisms, local symplectomorphisms, and symplectic embeddings. The different-dimensional definition here extends those finite-coordinate notions only by the displayed pullback equality.

(b) Equal-dimensional matrix and determinant tests. If $m_1=m_2=m$, then by 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 $X_H$ clause 1 the pullback condition is equivalent to

$DC(z{)}^{\mathsf{T}}{J}_{m}DC(z)=J_m\text{for every }z\in W_1\mathrm{.}$

The derivative is invertible at every point, so the map is a local diffeomorphism by The Euclidean inverse function theorem. Taking determinants of the matrix identity gives $(\det DC(z){)}^2=1$, hence $\mathrm{\mid }\det DC(z)\mathrm{\mid }=1$ for every $z$ (Invertible matrices and the general linear group ${\mathrm{GL}}_n(F)$, A positive-sized square matrix over a commutative ring is invertible if and only if its determinant is a unit, For $n\ge 1$, the determinant over a commutative ring by the Leibniz formula, and $\mathrm{\mid }\det A\mathrm{\mid }$ for a real matrix). In dimension $2m\ge 4$ the condition $\mathrm{\mid }\det DC\mathrm{\mid }=1$ alone is strictly weaker than canonicity (the companion page carries an explicit volume-preserving, non-canonical shear), while for $m=1$ the symplectic condition is exactly $\det DC=1$; absolute determinant $1$ alone also admits orientation-reversing, antisymplectic maps. If $m_1\mathrm{\ne }{m}_2$, $DC$ is rectangular, so no square determinant test is defined; the dimension-aware pullback identity of clause (a) is the criterion.

(c) Elementary closure and equal-dimensional field transport. By pointwise functoriality of pullback, the identity map, composites of composable canonical maps and inverses of canonical transformations are canonical. For a $C^2$ equal-dimensional canonical map $C:W_1\to W_2$ and a $C^2$ function $f$ on $W_2$, the Hamiltonian fields satisfy

$DC(z)X_{f\circ C}(z)=X_f(C(z))(z\in W_1),$

so that $C$ conjugates the Hamiltonian system of $f\circ C$ to that of $f$ by the chain rule and the equal-dimensional matrix identity: taking the inverse of $D{C}^{\mathsf{T}}{J}_{m}DC=J_m$ gives $DC{J}_{m}D{C}^{\mathsf{T}}=J_m$, which substituted into $DC{J}_m\mathrm{\nabla }(f\circ C)$ yields the displayed identity. No such ambient field-transport identity is asserted for a lower-dimensional symplectic immersion: the ambient field need not be tangent to its image. Bracket, solution and phase-volume consequences are likewise limited to their stated equal-dimensional local transformations.

(d) The model's canonical maps. For every generalized-coordinate immersion $X$, the momentum map ${\mathrm{\Phi }}_X$ in 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 a dimension-aware canonical map 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 ${\mathrm{\Phi }}_X^{\ast }\omega =\sum d{q}^a\wedge d{P}_a$; it is a symplectic immersion and is not asserted to be a global embedding. When $r=3N$, its local branches and the genuine transitions between such charts are equal-dimensional local canonical transformations by clauses 2-3 of that theorem. The drift, kick and velocity-Verlet maps of the mechanical splitting in exact arithmetic are equal-dimensional local canonical transformations on their admissible domains whenever those domains are nonempty, by The drift, kick and velocity-Verlet maps are canonical: $(DC{)}^{\mathsf{T}}JDC=J$ and $C^{\ast }\omega =\omega$.

(e) Status. The item defines the canonical maps of the presentation and fixes the dimension-aware pullback test of clause (a) and the equal-dimensional matrix test of clause (b). It asserts no existence of a canonical map beyond the listed suppliers, no global statement on multiply connected phase domains and no empirical content; in particular, canonicity is a local condition on the two-form, and an equal-dimensional canonical transformation need not admit a globally defined real generating function, as the scope remark of this page records.

The later The Hamiltonian evolution of a potential presentation preserves the two-form, the bracket and the determinant on its actual time-slice domains, also when the Hamiltonian depends on time and Generating functions for a potential presentation: first- and second-kind data generate canonical maps locally, the local converse under an invertible projection, and the time-dependent Hamiltonian $K=H+{\mathrm{\partial }}_{t}S$ establish further applications of this vocabulary to Hamiltonian time-slice maps and first- and second-kind generated maps, with their precise regularity and domain hypotheses. Those applications are conclusions of the later theorems, not prerequisites of this definition.