Cm Math Canonical

Let $n\ge 1$. A phase domain is a nonempty open set $U\subseteq {\mathbb{R}}^{2n}$ (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). Its points are written $z=(q,p)$ with $q=(q^1,\dots ,q^n)\in {\mathbb{R}}^n$ and $p=(p_1,\dots ,p_n)\in {\mathbb{R}}^n$; the coordinate functions $q^i,p_i:U\to \mathbb{R}$ are smooth ($C^k$ Euclidean maps and diffeomorphisms), and ${\mathrm{\partial }}_{q^1},\dots ,{\mathrm{\partial }}_{q^n},{\mathrm{\partial }}_{p_1},\dots ,{\mathrm{\partial }}_{p_n}$ is the coordinate frame (Directional derivatives and partial derivatives of a map $U\subseteq {\mathbb{R}}^m\to {\mathbb{R}}^n$). On a phase domain:

1. the canonical one-form is the smooth one-form ${\lambda }_U:={\sum }_{i=1}^n{p}_{i}d{q}^i\in {\mathrm{\Omega }}^1(U)$;
2. the standard symplectic two-form is the smooth two-form ${\omega }_U:={\sum }_{i=1}^{n}d{q}^i\wedge d{p}_i\in {\mathrm{\Omega }}^2(U)$ (A smooth differential $k$-form);
3. the standard symplectic matrix is the block matrix $J:=\left(\begin{array}{cc}{0}_n& I_n\\ -I_n& 0_n\end{array}\right)\in M_{2n}(\mathbb{R}),$ whose blocks are $n\times n$.

The identity ${\omega }_U=-d{\lambda }_U$ holds, because the coordinate formula for the exterior derivative (The local coordinate formula for the exterior derivative) gives $d{\lambda }_U={\sum }_{i}d{p}_i\wedge d{q}^i=-{\sum }_{i}d{q}^i\wedge d{p}_i$. When $U$ is an induced chart domain of a cotangent bundle, the published Tautological one-form on a cotangent bundle identifies ${\lambda }_U$ and ${\omega }_U$ with the tautological one-form and the canonical cotangent two-form in those coordinates; the present definition is the choice-free local coordinate model that the canonical-calculus items of this page use.

Let $U,V\subseteq {\mathbb{R}}^{2n}$ be phase domains with standard forms and matrix $J$ of A phase domain, its canonical one-form and its standard symplectic two-form. All finite differentiability classes and diffeomorphisms use $C^k$ Euclidean maps and diffeomorphisms.

On any open set $O\subseteq {\mathbb{R}}^m$, a $C^k$ coordinate differential form, $k\ge 0$, is a sum ${\sum }_{i_1<\cdots <i_j}{a}_{i_1\cdots i_j}(z)d{z}^{i_1}\wedge \cdots \wedge d{z}^{i_j}$ with $C^k$ coefficient functions. Its value at a point is the indicated alternating multilinear form on ${\mathbb{R}}^m$. For $k\ge 1$ define its exterior derivative by the coordinate formula $d(ad{z}^{i_1}\wedge \cdots )={\sum }_l({\mathrm{\partial }}_{l}a)d{z}^l\wedge d{z}^{i_1}\wedge \cdots$, extended linearly. For a $C^1$ map $C:U\to V$, the pointwise pullback is $(C^{\ast }\eta {)}_z(v_1,\dots ,v_j):={\eta }_{C(z)}(DC(z)v_1,\dots ,DC(z)v_j)\mathrm{.}$ It is a continuous form when $\eta$ is continuous; no assertion that this pullback is smooth is part of the definition. Wedges and contractions are the pointwise alternating multilinear operations. Thus the finite-regularity identities below can be read without applying a definition restricted to smooth forms or maps.

A $C^r$ canonical map, $1\le r\le \mathrm{\infty }$, is a $C^r$ map $C:U\to V$ satisfying $C^{\ast }{\omega }_V={\omega }_U$ pointwise. A $C^r$ canonical transformation is such a map that is a $C^r$ diffeomorphism onto $V$; a local $C^r$ canonical transformation is a $C^r$ local diffeomorphism satisfying the same equality. The $C^{\mathrm{\infty }}$ case agrees with Canonical transformation and Symplectomorphisms, local symplectomorphisms, and symplectic embeddings; the finite-$r$ terminology on this page is the definition just given, and does not assert that a $C^1$ map is smooth.

For $h\in C^2(U)$ the coordinate Hamiltonian field is the $C^1$ vector-valued map $X_h:=J\mathrm{\nabla }h$, viewed as a field by the canonical identification $T_{z}U={\mathbb{R}}^{2n}$. Its characterization by ${\iota }_{X_h}{\omega }_U=dh$ is proved in The Hamiltonian field of a $C^2$ function on a phase domain is $J\mathrm{\nabla }H$, and the Poisson bracket is ${\sum }_i({\mathrm{\partial }}_{q^i}f{\mathrm{\partial }}_{p_i}g-{\mathrm{\partial }}_{p_i}f{\mathrm{\partial }}_{q^i}g)$ ↗. For $f,g\in C^1(U)$ define the coordinate Poisson bracket by $\{f,g\}:=\mathrm{\nabla }{f}^{\mathsf{T}}J\mathrm{\nabla }g$; for $C^2$ inputs it is $C^1$ and equals ${\omega }_U(X_f,X_g)$ by that proposition. Smooth inputs recover the smooth symplectic conventions. No closure of the $C^2$ class under this bracket is assumed.

Let $n\ge 1$, let $U$ be a phase domain with canonical one-form ${\lambda }_U$, standard two-form ${\omega }_U={\sum }_{i}d{q}^i\wedge d{p}_i$ and matrix $J=\left(\begin{array}{cc}{0}_n& I_n\\ -I_n& 0_n\end{array}\right)$ (A phase domain, its canonical one-form and its standard symplectic two-form). For a point $z\in U$ write ${\omega }_z:=({\omega }_U{)}_z$. Then:

1. ${\omega }_U$ is closed: $d{\omega }_U=0$;
2. for all $v,w\in {\mathbb{R}}^{2n}$, written in the coordinate frame as $v=(a,b)$ and $w=(c,d)$ with $a,b,c,d\in {\mathbb{R}}^n$, one has ${\omega }_z(v,w)=v^{\mathsf{T}}Jw=a\cdot d-b\cdot c;$
3. for every $v=(a,b)\in {\mathbb{R}}^n\times {\mathbb{R}}^n$ the contraction is the covector ${\iota }_v{\omega }_z={\sum }_{i=1}^n{a}_{i}d{p}_i-{\sum }_{i=1}^n{b}_{i}d{q}^i;$
4. $J^{\mathsf{T}}=-J$, $J^2=-I_{2n}$, and $J$ is invertible with $J^{-1}=J^{\mathsf{T}}=-J$;
5. consequently $v\mapsto {\iota }_v{\omega }_z$ is a linear isomorphism ${\mathbb{R}}^{2n}\to ({\mathbb{R}}^{2n}{)}^{\ast }$, so ${\omega }_U$ is nondegenerate at every point and $(U,{\omega }_U)$ is a symplectic manifold.

• The point of the block form is that all five clauses are pointwise linear-algebra statements with constant coefficients; no completeness, boundedness or topological hypothesis on $U$ is used.
• The published The canonical cotangent two-form is symplectic states the same closedness and nondegeneracy for the canonical form on a cotangent bundle (under countable choice); the present lemma is the choice-free local form on a phase domain, stated so that the flow, canonical-map and generating-function items below can compute with it directly.

Let $n\ge 1$, let $U$ be a phase domain with standard two-form ${\omega }_U={\sum }_{i}d{q}^i\wedge d{p}_i$ (A phase domain, its canonical one-form and its standard symplectic two-form), and let $H:U\to \mathbb{R}$ be of class $C^2$ ($C^k$ Euclidean maps and diffeomorphisms). Write $\mathrm{\nabla }H=(H_{q^1},\dots ,H_{q^n},H_{p_1},\dots ,H_{p_n})$ for the Euclidean gradient (The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case) and $J$ for the standard symplectic matrix. Then:

1. there is exactly one $C^1$ vector field $X_H$ on $U$ with ${\iota }_{X_H}{\omega }_U=dH$, and it is $X_H=J\mathrm{\nabla }H=(H_{p_1},\dots ,H_{p_n},-H_{q^1},\dots ,-H_{q^n});$
2. an integral curve $t\mapsto (q(t),p(t))$ of $X_H$ satisfies Hamilton's equations ${\dot{q}}^i=H_{p_i}(q,p),{\dot{p}}_i=-H_{q^i}(q,p)(i=1,\dots ,n);$
3. for $f,g\in C^2(U)$ the Poisson bracket $\{f,g\}:={\omega }_U(X_f,X_g)$ (in the finite-regularity convention of Finite-regularity differential forms, Hamiltonian fields and canonical maps in phase charts) is $\{f,g\}=\mathrm{\nabla }{f}^{\mathsf{T}}J\mathrm{\nabla }g={\sum }_{i=1}^n(f_{q^i}{g}_{p_i}-f_{p_i}{g}_{q^i}),$ hence $\{f,g\}=-\{g,f\}$, and $\{f,g\}$ is $C^1$ on $U$.

• The published Hamiltonian vector fields exist uniquely for smooth functions and Hamilton equations in canonical cotangent coordinates give existence, uniqueness and the Hamilton equations for smooth functions on a cotangent bundle (the latter under countable choice). The point of this item is the choice-free local statement for $C^2$ data on a phase domain, together with the explicit block form of the field that the flow and canonical-map items below differentiate.
• Part 3 is the coordinate formula on which the published Coordinate formula for the Poisson bracket and the bracket identities of the root library (bilinearity, skew symmetry, the derivation rule and the Jacobi identity in The Poisson bracket is bilinear, skew, and a derivation in each entry and The Poisson bracket satisfies the Jacobi identity) can be invoked by consumers in the smooth subcase; no second proof of the Jacobi identity is developed on this page.

Let $n\ge 1$, let $U$ be a phase domain with standard two-form ${\omega }_U$ and matrix $J$, let $H\in C^2(U)$, and let $\mathrm{\Phi }$ be the local flow of the Hamiltonian field $X_H=J\mathrm{\nabla }H$ (The Hamiltonian field of a $C^2$ function on a phase domain is $J\mathrm{\nabla }H$, and the Poisson bracket is ${\sum }_i({\mathrm{\partial }}_{q^i}f{\mathrm{\partial }}_{p_i}g-{\mathrm{\partial }}_{p_i}f{\mathrm{\partial }}_{q^i}g)$, The local flow of an autonomous first-order system). Let $z_0\in U$, let $I\subseteq \mathbb{R}$ be the open interval of times $t$ with $(t,z_0)$ in the flow domain, and for $t\in I$ put

$Y(t):=D_z{\mathrm{\Phi }}_t(z_0)\in M_{2n}(\mathbb{R})\mathrm{.}$

Then $t\mapsto Y(t)$ is of class $C^1$ on $I$ and, for every $t\in I$:

1. (variational equation) $Y^{\mathrm{\prime }}(t)=J\mathrm{H}\mathrm{e}\mathrm{s}\mathrm{s}H({\mathrm{\Phi }}_t(z_0))Y(t)$ and $Y(0)=I_{2n}$;
2. (canonical matrix identity) $Y(t{)}^{\mathsf{T}}JY(t)=J$;
3. (unit determinant) $\det Y(t)=1$.

The same conclusions hold for a time-dependent Hamiltonian $\mathcal{H}:T\times U\to \mathbb{R}$ of class $C^2$, where $T$ is an open real interval. For $t_0\in T$ and $z_0\in U$, write $E(t,t_0,z)$ for the maximal solution of $z^{\mathrm{\prime }}=J{\mathrm{\nabla }}_z\mathcal{H}(t,z)$ with $z(t_0)=z$; its domain consists of the triples where this solution exists. On the maximal open time interval $I_{t_0,z_0}$ set $Y(t)=D_{z}E(t,t_0,z_0)$. Then $Y^{\mathrm{\prime }}=J{\mathrm{Hess}}_z\mathcal{H}(t,E(t,t_0,z_0))Y,Y(t_0)=I_{2n},Y^{\mathsf{T}}JY=J,\det Y=1.$ The evolution $E$ is $C^1$ on its open domain, and each fixed-endpoint map has inverse $E(t_0,t,\cdot )$ on its image. Autonomous $H$ is recovered by $\mathcal{H}(t,z)=H(z)$ and $E(t,t_0,z)=\mathrm{\Phi }(t-t_0,z)$.

• Clauses 2 and 3 are the differentiated form of the published statement that Hamiltonian flows preserve the symplectic form (Hamiltonian flows preserve the symplectic form) and the Liouville volume (Liouville volume preservation); the present item records the matrix identity that a chart computation, rather than the Cartan formula, produces, and it is this identity that the canonical-map test of 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 $X_H$ compares with.
• The statement is local in $t$ and on the actual flow domain; no completeness of $X_H$ is assumed, exactly as in The local flow of an autonomous first-order system.

Assume the Axiom of Countable Choice where the cited change-of-variables theorem requires it (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), A C^1 diffeomorphism satisfies the change-of-variables formula for L^1 functions). Let $n\ge 1$, let $U$ be a phase domain (A phase domain, its canonical one-form and its standard symplectic two-form), let $H\in C^2(U)$, and let $\mathrm{\Phi }$ be the local flow of $X_H=J\mathrm{\nabla }H$ (The Hamiltonian field of a $C^2$ function on a phase domain is $J\mathrm{\nabla }H$, and the Poisson bracket is ${\sum }_i({\mathrm{\partial }}_{q^i}f{\mathrm{\partial }}_{p_i}g-{\mathrm{\partial }}_{p_i}f{\mathrm{\partial }}_{q^i}g)$). Let $t\in \mathbb{R}$, let $D_t:=\{z\in U:(t,z)\text{ lies in the flow domain}\}$ be the open domain of ${\mathrm{\Phi }}_t$ (The flow domain of a C¹ autonomous field is open and the flow is C¹ on it), and let $A\subseteq D_t$ be a bounded Borel set with $\bar{A}\subseteq D_t$. Then $A$ and ${\mathrm{\Phi }}_t(A)$ are Lebesgue measurable with

${\lambda }_{2n}({\mathrm{\Phi }}_t(A))={\lambda }_{2n}(A)\mathrm{.}$

For a $C^2$ time-dependent Hamiltonian $\mathcal{H}:T\times U\to \mathbb{R}$ and the evolution $E$ of The derivative of the Hamiltonian flow satisfies $Y^{\mathsf{T}}JY=J$ and $\det Y=1$, fix $t_0,t\in T$. Its initial-state domain $D_{t,t_0}=\{z:E(t,t_0,z)\text{ is defined}\}$ is open. For every bounded Borel $A\subseteq D_{t,t_0}$ with $\bar{A}\subseteq D_{t,t_0}$, ${\lambda }_{2n}(E(t,t_0,A))={\lambda }_{2n}(A)\mathrm{.}$ No completeness or time independence is needed.