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The linear shear: the second-kind generating function $S (q, P) = q P + λ q^{2} / 2$ and the matrix test

Example

Let $n = 1$ , let $λ \in R$ , and let $S : R^{2} \to R$ , $S (q, P) : = q P + λ q^{2} / 2$ . Then:

$S$ is a generating function of the second kind with $S_{q P} = 1 \neq 0$ , and the local map it generates is $Ψ_{λ} (q, p) = (Q, P) = (q, p - λ q)$ , defined on all of $R^{2}$ (Generating functions of the first and second kind for a local canonical map);
$Ψ_{λ}$ is canonical, both by the generating-function theorem (The local map generated by a first- or second-kind generating function is a canonical $C^{2}$ local diffeomorphism) and by direct verification of the matrix identity $D Ψ_{λ}^{T} J D Ψ_{λ} = J$ (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 $X_{H}$ ).

Facts & Assumptions

Given: The real number $λ$ and the function $S (q, P) = q P + λ q^{2} / 2$ .

[F1]

Derivatives (Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ ): $S_{q} = P + λ q$ and $S_{P} = q$ , so $S_{q P} = 1$ and the second-kind relations $p = S_{q}$ , $Q = S_{P}$ read $p = P + λ q$ , $Q = q$ ; solving, $P = p - λ q$ and $Q = q$ , so the generated map is $Ψ_{λ} (q, p) = (q, p - λ q)$ on all of $R^{2}$ , a $C^{\infty}$ diffeomorphism (Generating functions of the first and second kind for a local canonical map, $C^{k}$ Euclidean maps and diffeomorphisms).

[F2]

Matrix test (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 $X_{H}$ , The standard two-form of a phase domain is closed and nondegenerate, with contraction $ι_{v} ω = \sum_{i} a_{i} d p_{i} - \sum_{i} b_{i} d q^{i}$ , Rectangular matrix multiplication and the identity matrix $I_{n}$ , including zero-sized shapes): a $C^{1}$ map $Ψ$ between phase domains is canonical iff $D Ψ^{T} J D Ψ = J$ ; here $J = (\begin{array}{cc} 0 & 1 \\ - 1 & 0 \end{array})$ .

Verification

technique · direct; read off the generated map from the defining relations and multiply the two-by-two matrices

1.1F1algebra

Clause 1 is [F1]: $S_{q P} = 1 \neq 0$ , and the relations $p = S_{q} = P + λ q$ , $Q = S_{P} = q$ give $Ψ_{λ} (q, p) = (q, p - λ q)$ , which is a linear isomorphism of $R^{2}$ and therefore a $C^{\infty}$ diffeomorphism.

2.1F1F2step 1.1algebra∎

Clause 2: the derivative is $D Ψ_{λ} = (\begin{array}{cc} 1 & 0 \\ - λ & 1 \end{array})$ , and $(\begin{array}{cc} 1 & - λ \\ 0 & 1 \end{array}) (\begin{array}{cc} 0 & 1 \\ - 1 & 0 \end{array}) (\begin{array}{cc} 1 & 0 \\ - λ & 1 \end{array}) = (\begin{array}{cc} 0 & 1 \\ - 1 & 0 \end{array}) = J$ by direct block multiplication; hence $Ψ_{λ}$ is canonical by [F2], while the generating-function theorem gives the same conclusion because $d S = p d q + Q d P$ and $d^{2} S = 0$ force $d q \land d p = d Q \land d P$ .

The linear shear: the second-kind generating function $S (q, P) = q P + λ q^{2} / 2$ and the matrix test

Example

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Sources

The linear shear: the second-kind generating function S(q,P)=qP+λq2/2S(q,P)=qP+λq2/2 and the matrix test

Example

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Sources

The linear shear: the second-kind generating function $S (q, P) = q P + λ q^{2} / 2$ and the matrix test