The Poisson bracket of phase functions of a potential presentation: $\{f,g\}=\omega (X_f,X_g)$ and the coordinate formula in each canonical chart

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. Dimensions are read by Dimensions and SI unit conventions of the non-relativistic classical mechanics framework; differentiability classes are those of $C^k$ Euclidean maps and diffeomorphisms.

(a) Phase functions. Let $W_0$ be a nonempty open subset of a phase domain of the presentation. A phase function of the presentation on $W_0$ of class $C^k$ is a map $f:W_0\to \mathbb{R}$ of class $C^k$ ($k\ge 1$); its physical reading is a model quantity computed from the positions and momenta of the labels, and its dimension is declared as part of the datum. A time-dependent phase function on an interval $J\subseteq I_0$ is a map $f:J\times W_0\to \mathbb{R}$ of the declared class, written $f_t:=f(t,\cdot )$.

(b) The bracket. For $C^1$ phase functions $f,g:W_0\to \mathbb{R}$, define their Poisson bracket for the presentation by the pointwise coordinate formula $\{f,g\}=\mathrm{\nabla }{f}^{\mathsf{T}}J\mathrm{\nabla }g$ in a canonical chart, in the finite-regularity convention of Finite-regularity differential forms, Hamiltonian fields and canonical maps in phase charts. This is a continuous function. The definition is independent of the canonical chart: for a canonical $C^1$ coordinate change with derivative $M$, the matrix identity implies $MJ{M}^{\mathsf{T}}=J$, and the equivalence of the two matrix identities is proved in 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 $X_H$, and the chain rule transforms the two gradients so their pairing is unchanged.

When both functions are $C^2$, their 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 are $C^1$, and the bracket has the intrinsic formula $\{f,g\}=\omega (X_f,X_g)=df(X_g)=X_g(f)=-X_f(g)\mathrm{.}$ In this case it is $C^1$, with the same sign convention as the smooth root bracket and 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 extension to $C^1$ inputs makes evolution identities for $C^1$ observables and nested brackets of $C^2$ functions meaningful; it does not assert closure of the $C^2$ class under the bracket. Its dimension is $[f][g](\mu {\mathrm{\ell }}^2{\tau }^{-1}{)}^{-1}$, the product of the entries' dimensions divided by the action dimension of the two-form.

(c) Coordinate formula. In the Cartesian canonical coordinates $(x,p)$ of the presentation, and more generally in any canonical chart $({\theta }^1,\dots ,{\theta }^{2m})$ 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,

$\{f,g\}={\sum }_{a=1}^{3N}(f_{x^a}{g}_{p_a}-f_{p_a}{g}_{x^a})=\mathrm{\nabla }{f}^{\mathsf{T}}J\mathrm{\nabla }g,\{f,g\}={\sum }_{a=1}^m(f_{{\theta }^a}{g}_{{\theta }^{m+a}}-f_{{\theta }^{m+a}}{g}_{{\theta }^a}),$

with derivatives those of The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case and Directional derivatives and partial derivatives of a map $U\subseteq {\mathbb{R}}^m\to {\mathbb{R}}^n$. The coordinate formula is recorded here and its identities (bilinearity, skew-symmetry, the Leibniz rule, the Jacobi identity and the fundamental brackets) are proved in the bracket-identities theorem of this page.

(d) Status. The item defines a bilinear operation on the declared phase domain by the pointwise coordinate formula, and for $C^2$ entries by $\{f,g\}=\omega (X_f,X_g)$, hence independently of the canonical chart used to evaluate the coordinate formula. It asserts no equation of motion, no conservation law, no algebraic identity beyond well-definedness and no measured value; in particular the bracket with the Hamiltonian and the notion of first integral are the subject of the evolution theorem of this page.