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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 $X_{H}$

Statement

Let $n \geq 1$ , let $U, V \subseteq R^{2 n}$ be phase domains with standard two-forms $ω_{U}$ , $ω_{V}$ and the same matrix $J$ (A phase domain, its canonical one-form and its standard symplectic two-form), and let $C : U \to V$ be of class $C^{1}$ ( $C^{k}$ Euclidean maps and diffeomorphisms). Then:

$C^{*} ω_{V} = ω_{U}$ if and only if $D C (z)^{T} J D C (z) = J$ for every $z \in U$ ;
if $C$ is a $C^{1}$ diffeomorphism onto $V$ and $C^{*} ω_{V} = ω_{U}$ , then $C$ is a $C^{1}$ canonical transformation in the finite-regularity sense of Finite-regularity differential forms, Hamiltonian fields and canonical maps in phase charts and $∣ \det D C (z) ∣ = 1$ for every $z \in U$ ;
if in addition $H \in C^{2} (V)$ and $C : U \to V$ is a $C^{2}$ diffeomorphism with $C^{*} ω_{V} = ω_{U}$ , then $H \circ C \in C^{2} (U)$ and $D C (z) X_{H \circ C} (z) = X_{H} (C (z)) (z \in U) .$

Facts & Assumptions

Given: Phase domains $U, V$ , a $C^{1}$ map $C : U \to V$ , and in clauses 2 and 3 the corresponding additional regularity and bijectivity hypotheses.

[F1]

Pullback and evaluation: for a two-form $τ$ and $z \in U$ , $(C^{*} τ)_{z} (v, w) = τ_{C (z)} (D C (z) v, D C (z) w)$ (Finite-regularity differential forms, Hamiltonian fields and canonical maps in phase charts, The total (Fréchet) derivative $D f (a)$ as the linear first-order approximation with $o (∥ h ∥_{2})$ remainder).

[F2]

Evaluation and block identities of the standard form: for $ω_{V}$ at a point $y \in V$ and vectors $ξ, η \in R^{2 n}$ one has $(ω_{V})_{y} (ξ, η) = ξ^{T} J η$ , and $J^{T} = - J$ , $J^{2} = - I_{2 n}$ , $J^{- 1} = J^{T} = - J$ (The standard two-form of a phase domain is closed and nondegenerate, with contraction $ι_{v} ω = \sum_{i} a_{i} d p_{i} - \sum_{i} b_{i} d q^{i}$ ).

[F3]

Calculus and linear algebra: $X_{h} = J \nabla h$ for $C^{2}$ functions $h$ (The Hamiltonian field of a $C^{2}$ function on a phase domain is $J \nabla H$ , and the Poisson bracket is $\sum_{i} (\partial_{q^{i}} f \partial_{p_{i}} g - \partial_{p_{i}} f \partial_{q^{i}} g)$ ); the chain rule gives $\nabla (H \circ C) (z) = D C (z)^{T} \nabla H (C (z))$ (The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ , The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case). For invertible $M : = D C$ , the identity $M^{T} J M = J$ is equivalent to $M J M^{T} = J$ (each direction follows by the same inversion argument) and implies $(\det M)^{2} = 1$ : inverting gives $M^{- 1} J^{- 1} (M^{- 1})^{T} = J^{- 1}$ , and $J^{- 1} = J^{T} = - J$ from [F2] turns this into $M^{- 1} J (M^{- 1})^{T} = J$ , which after multiplying on the left by $M$ and on the right by $M^{T}$ is $M J M^{T} = J$ ; taking determinants and using $\det (A B) = \det (A) \det (B)$ , $\det (A^{T}) = \det (A)$ and $\det J \neq 0$ gives $(\det M)^{2} \det J = \det J$ (The standard two-form of a phase domain is closed and nondegenerate, with contraction $ι_{v} ω = \sum_{i} a_{i} d p_{i} - \sum_{i} b_{i} d q^{i}$ , Invertible Euclidean linear maps, Rectangular matrix multiplication and the identity matrix $I_{n}$ , including zero-sized shapes, For same-sized finite square matrices over a commutative ring, $\det (A B) = \det (A) \det (B)$ , For every square matrix over a commutative ring, $\det (A^{T}) = \det (A)$ , A positive-sized square matrix over a commutative ring is invertible if and only if its determinant is a unit, For $n \geq 1$ , the determinant over a commutative ring by the Leibniz formula, and $∣ \det A ∣$ for a real matrix, Linear map between vector spaces over the same field).

Proof

technique · direct; evaluate the pullback on a pair of vectors to obtain the matrix criterion, then differentiate $H\circ C$ and use the transposed matrix identity

1.1F1F2algebra

Clause 1: by [F1], for all $z$ , $v$ and $w$ , $(C^{*} ω_{V})_{z} (v, w) = (D C (z) v)^{T} J (D C (z) w) = v^{T} (D C (z)^{T} J D C (z)) w$ by [F1] and [F2]; since $ω_{U}$ is evaluated as $v^{T} J w$ by [F2], and since two bilinear forms are equal iff their matrices in the coordinate basis agree, $C^{*} ω_{V} = ω_{U}$ is equivalent to $D C (z)^{T} J D C (z) = J$ for every $z \in U$ .

2.1step 1.1F3algebra

Clause 2: a $C^{1}$ diffeomorphism with $C^{*} ω_{V} = ω_{U}$ is by Finite-regularity differential forms, Hamiltonian fields and canonical maps in phase charts a $C^{1}$ canonical transformation; taking determinants in the matrix identity of clause 1 and using the determinant identities of [F3] gives $(\det D C)^{2} = 1$ , hence $∣ \det D C ∣ = 1$ .

3.1F3step 1.1algebra∎

Clause 3: assume $H \in C^{2} (V)$ and $C$ a $C^{2}$ diffeomorphism with $C^{*} ω_{V} = ω_{U}$ . Then $H \circ C \in C^{2} (U)$ and, by [F3], $X_{H \circ C} (z) = J \nabla (H \circ C) (z) = J D C (z)^{T} \nabla H (C (z))$ ; the identity $D C^{T} J D C = J$ of clause 1, with $D C$ invertible, gives $D C J D C^{T} = J$ by the equivalence proved in [F3]; hence $D C (z) X_{H \circ C} (z) = D C (z) J D C (z)^{T} \nabla H (C (z)) = J \nabla H (C (z)) = X_{H} (C (z))$ , which is the asserted transport identity.

Remarks

Clause 1 is the pointwise coordinate test of Finite-regularity differential forms, Hamiltonian fields and canonical maps in phase charts; in the smooth subcase the published Symplectomorphisms, local symplectomorphisms, and symplectic embeddings and Canonical transformation define the same notion intrinsically, and clause 1 supplies the determinant-like matrix test used by the generating-function items below.
Clause 3 says that the Hamiltonian-field assignment is natural under canonical maps; it is the precise sense in which "canonical transformations preserve Hamilton's equations" and is used in the time-dependent generating-function item.

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 $X_{H}$

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources

A C1C1 map between phase domains preserves the standard two-form iff DCTJ DC=JDCTJDC=J, and a canonical diffeomorphism carries XH∘CXH∘C​ to XHXH​

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources

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 $X_{H}$