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A canonical map with invertible $(q, Q)$ -projection is locally generated by a first-kind generating function

Statement

Let $n \geq 1$ , let $U, V \subseteq R^{2 n}$ be phase domains with standard two-forms $ω_{U}, ω_{V}$ , and let $C : U \to V$ be a $C^{2}$ diffeomorphism with $C^{*} ω_{V} = ω_{U}$ (A phase domain, its canonical one-form and its standard symplectic two-form, A $C^{1}$ map between phase domains preserves the standard two-form iff $D C^{T} J D C = J$ , and a canonical diffeomorphism carries $X_{H \circ C}$ to $X_{H}$ ). Write $C (z) = (Q (z), P (z))$ and $z = (q (z), p (z))$ for the coordinate functions, and let $z_{0} \in U$ . If the map $F : U \to R^{n} \times R^{n}, F (z) : = (q (z), Q (z))$ has invertible derivative at $z_{0}$ , then there are an open neighbourhood $W \subseteq U$ of $z_{0}$ and an open ball $W^{''} \subseteq R^{n} \times R^{n}$ about $F (z_{0})$ with $F (W) = W^{''}$ on which $F$ is a $C^{2}$ diffeomorphism, and a $C^{3}$ function $S : W^{''} \to R$ with $\det S_{q Q} (F (z_{0})) \neq 0$ , such that the restriction of $S$ to a product neighbourhood $O_{q} \times O_{Q} \subseteq W^{''}$ of $F (z_{0})$ is a first-kind generating function and its generated local map agrees with $C$ on a neighbourhood of $z_{0}$ .

The words canonical and the finite-regularity form operations here use Finite-regularity differential forms, Hamiltonian fields and canonical maps in phase charts, with the displayed $C^{2}$ maps; smoothness is not inferred.

Facts & Assumptions

Given: The canonical $C^{2}$ diffeomorphism $C$ , the point $z_{0}$ , and the projection $F$ .

[F1]

Since $C$ is $C^{2}$ , so is $F$ , with block derivative $D F (z_{0}) = (\begin{array}{cc} I_{n} & 0 \\ Q_{q} (z_{0}) & Q_{p} (z_{0}) \end{array})$ ; invertibility of $D F (z_{0})$ is therefore equivalent to $\det Q_{p} (z_{0}) \neq 0$ , the determinant being that of For $n \geq 1$ , the determinant over a commutative ring by the Leibniz formula, and $∣ \det A ∣$ for a real matrix (The total (Fréchet) derivative $D f (a)$ as the linear first-order approximation with $o (∥ h ∥_{2})$ remainder, The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case, Invertible Euclidean linear maps, Rectangular matrix multiplication and the identity matrix $I_{n}$ , including zero-sized shapes).

[F2]

Euclidean inverse function theorem (The Euclidean inverse function theorem): a $C^{2}$ local diffeomorphism at $z_{0}$ has a $C^{2}$ local inverse; hence there are open neighbourhoods $W$ of $z_{0}$ and $W^{''}$ of $F (z_{0})$ such that $F : W \to W^{''}$ is a $C^{2}$ diffeomorphism, and the inverse $G : W^{''} \to W$ , $(q, Q) \mapsto z$ , is $C^{2}$ .

[F3]

Pullback and canonicity in coordinates (The pullback of a differential form, The wedge product of differential forms, A $C^{1}$ map between phase domains preserves the standard two-form iff $D C^{T} J D C = J$ , and a canonical diffeomorphism carries $X_{H \circ C}$ to $X_{H}$ ): $C^{*} ω_{V} = ω_{U}$ reads $\sum_{i} d Q^{i} \land d P_{i} = \sum_{i} d q^{i} \land d p_{i}$ as forms on $U$ .

[F4]

Poincare lemma for one-forms (Poincare's lemma on a star-shaped domain: every closed C1 field is exact, Star-shaped open subsets of Euclidean space): a closed $C^{1}$ one-form on a star-shaped open subset of $R^{m}$ is exact, with a $C^{2}$ potential given by the radial integral; the potential of a $C^{2}$ closed form is $C^{3}$ because its gradient is that $C^{2}$ form ( $C^{k}$ Euclidean maps and diffeomorphisms, Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ ).

[F5]

First-kind generating functions and the exterior derivative (Generating functions of the first and second kind for a local canonical map, The local coordinate formula for the exterior derivative, The exterior derivative squares to zero, Closed and exact differential forms): $S$ is a first-kind generating function when $\det S_{q Q} \neq 0$ , and then $p = S_{q}$ , $P = - S_{Q}$ ; conversely a function with those two properties and nonvanishing mixed determinant satisfies the definition.

Proof

technique · direct; invert the projection to coordinates $(q,Q)$, show that the one-form $p\,dq-P\,dQ$ is closed by canonicity, and integrate it on a star-shaped neighbourhood

1.1F1F2

By [F1] and [F2], $F : W \to W^{''}$ is a $C^{2}$ diffeomorphism with $C^{2}$ inverse $G$ ; on $W^{''}$ the functions $p_{i} : = p_{i} \circ G$ and $P_{i} : = P_{i} \circ G$ are $C^{2}$ in $(q, Q)$ .

2.1step 1.1F3algebra

Define the $C^{2}$ one-form $α : = \sum_{i} p_{i} d q^{i} - \sum_{i} P_{i} d Q^{i}$ on $W^{''}$ , with the functions of step 1.1. Its exterior derivative is $d α = \sum_{i} d p_{i} \land d q^{i} - \sum_{i} d P_{i} \land d Q^{i} = \sum_{i} d Q^{i} \land d P_{i} - \sum_{i} d q^{i} \land d p_{i}$ (using $d p \land d q = - d q \land d p$ and $d P \land d Q = - d Q \land d P$ ); by [F3] this vanishes, so $α$ is closed (Closed and exact differential forms).

3.1step 2.1F4

The ball $W^{''}$ is open and star-shaped about its centre $F (z_{0})$ (Star-shaped open subsets of Euclidean space, 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), and the one-form $α$ of step 2.1 is a closed $C^{1}$ one-form on it (indeed $C^{2}$ ); by [F4] there is a $C^{3}$ function $S : W^{''} \to R$ with $d S = α$ , that is, $S_{q^{i}} = p_{i}$ and $S_{Q^{j}} = - P_{j}$ .

4.1F1step 3.1algebra

At $F (z_{0})$ one has $\det S_{q Q} = \det (\partial p_{i} / \partial Q^{j})$ ; since $D G$ is the inverse of $D F$ and $D G$ has block form $(\begin{array}{cc} I_{n} & 0 \\ \partial p / \partial q & \partial p / \partial Q \end{array})$ , the identity $D G D F = I_{2 n}$ gives $(\partial p / \partial Q) (z_{0}) (Q_{p} (z_{0})) = I_{n}$ ; by [F1] the factor $Q_{p} (z_{0})$ is invertible, hence $\det S_{q Q} (F (z_{0})) \neq 0$ .

5.1step 3.1step 4.1F5∎

By step 4.1 and continuity of the mixed determinant, choose a product neighbourhood $O_{q} \times O_{Q} \subseteq W^{''}$ of $F (z_{0})$ on which $\det S_{q Q} \neq 0$ . Its restriction is precisely first-kind data in [F5]; the relations $p = S_{q}$ , $P = - S_{Q}$ of step 3.1 are exactly the defining relations of the map generated by $S$ (Generating functions of the first and second kind for a local canonical map), and they hold on the image of $C$ under the identification $F$ ; hence the generated local map agrees with $C$ near $z_{0}$ .

Remarks

The hypothesis that $F (z) = (q (z), Q (z))$ have invertible derivative at $z_{0}$ is the classical no-caustic condition for the first-kind description; when it fails one uses a second-kind or mixed generating function, and globally one may meet genuine obstructions such as the cylinder shift of The symplectic shift of the cylinder changes $λ$ by the closed, non-exact form $a d θ$ and has no global generating function.
The published Liouville–Arnold action–angle theorem uses generating functions only through its action-angle construction; no global generating function is asserted here.

A canonical map with invertible $(q, Q)$ -projection is locally generated by a first-kind generating function

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources

A canonical map with invertible (q,Q)(q,Q)-projection is locally generated by a first-kind generating function

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources

A canonical map with invertible $(q, Q)$ -projection is locally generated by a first-kind generating function