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

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 data and conventions of The momentum of a potential presentation: the fibre derivative of the mechanical Lagrangian, the Cartesian momentum Mv and generalized momenta: masses $m_1,\dots ,m_N>0$ and nonempty open configuration domain $Q\subseteq {\mathbb{R}}^{3N}$ (Configuration domain of $N$ labeled particles: open admissible sets, regular holonomic constraints and the mass operator), time window $I_0$, mass operator $\mathsf{M}$, a $C^2$ potential $U:I_0\times Q\to \mathbb{R}$ of dimension $\mu {\mathrm{\ell }}^2{\tau }^{-2}$ and residual forces $d$ of the standing regularity (Potential presentations of a force model, potential and mechanical energy, closed versus exact force fields, local potentials and equipotential level sets, Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance). Dimensions are read by Dimensions and SI unit conventions of the non-relativistic classical mechanics framework, the Euclidean product on ${\mathbb{R}}^{3N}$ is The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$, and differentiability classes are those of $C^k$ Euclidean maps and diffeomorphisms. Write $n:=3N\ge 1$. The Cartesian momentum phase chart of the presentation is $I_0\times Q\times {\mathbb{R}}^{3N}$ with the momentum $p=\mathsf{M}v$ of The momentum of a potential presentation: the fibre derivative of the mechanical Lagrangian, the Cartesian momentum Mv and generalized momenta, and the momentum phase chart of a generalized-coordinate map $X:\mathrm{\Omega }\to Q$ with $r$ coordinates is $I_0\times \mathrm{\Omega }\times {\mathbb{R}}^r$ with the generalized momentum $P=G(q)v$, $G(q)=DX(q{)}^{\mathsf{T}}\mathsf{M}DX(q)$ (Generalized-coordinate maps of the model, the substituted Lagrangian, the kinetic metric and generalized forces, Hyperregular potential presentations and their inverse Legendre branches in the model's charts).

(a) Cartesian chart. The Cartesian phase domain of the presentation is the nonempty open set

$W:=Q\times {\mathbb{R}}^{3N}\subseteq {\mathbb{R}}^{2n},z=(x,p),$

a phase domain in the sense of A phase domain, its canonical one-form and its standard symplectic two-form (openness is clause (a) of Configuration domain of $N$ labeled particles: open admissible sets, regular holonomic constraints and the mass operator; nonemptiness is an explicit hypothesis here, so an empty configuration domain has no instances of this phase-domain definition). On $W$ the canonical one-form of the presentation in the Cartesian chart is the smooth one-form

${\lambda }_{(x,p)}(\delta x,\delta p):=p\cdot \delta x((x,p)\in W,(\delta x,\delta p)\in {\mathbb{R}}^{3N}\times {\mathbb{R}}^{3N}),$

where $p\cdot \delta x$ is the Euclidean pairing of the momentum covector with the displacement, and the two-form of the presentation in the Cartesian chart is

$\omega :=-d\lambda$

(A smooth differential $k$-form). In coordinates, reading $x=(x^1,\dots ,x^{3N})$ and $p=(p_1,\dots ,p_{3N})$ as the standard coordinates of $W$,

$\lambda ={\sum }_{a=1}^{3N}{p}_{a}d{x}^a,\omega ={\sum }_{a=1}^{3N}d{x}^a\wedge d{p}_a,{\omega }_{(x,p)}((\delta x,\delta p),(\delta x^{\mathrm{\prime }},\delta p^{\mathrm{\prime }}))=\delta x\cdot \delta p^{\mathrm{\prime }}-\delta p\cdot \delta x^{\mathrm{\prime }},$

the last identity using the Euclidean product. The dimension of both forms is that of action, $\mu {\mathrm{\ell }}^2{\tau }^{-1}$: the pairing of a momentum ($\mu \mathrm{\ell }{\tau }^{-1}$) with a displacement ($\mathrm{\ell }$). The forms have no explicit time dependence; for each $t\in I_0$ the slice $\{t\}\times W$ carries the same two-form. The momentum phase chart with time, $I_0\times W$ with coordinates $(t,x,p)$, is the domain of the Hamiltonians and fields of this page. The sign convention fixed by Hamiltonian vector field and Hamiltonian function and used throughout is ${\iota }_X\omega =dH$; $H$ denotes a phase function as in that item.

(b) Generalized-coordinate chart. Let $X:\mathrm{\Omega }\to Q$ be a generalized-coordinate map with substituted Lagrangian $\tilde{L}$ and kinetic metric $G$, and let $r$ be its number of coordinates (Generalized-coordinate maps of the model, the substituted Lagrangian, the kinetic metric and generalized forces). The chart map of the presentation in the coordinates $X$ is

${\mathrm{\Phi }}_X:\mathrm{\Omega }\times {\mathbb{R}}^r⟶Q\times {\mathbb{R}}^{3N},{\mathrm{\Phi }}_X(q,P):=(X(q),\mathsf{M}DX(q)G(q{)}^{-1}P)\mathrm{.}$

Here $DX(q)$ is the $(3N)\times r$ Jacobian of $X$ at $q$ (The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case), $G(q)=DX(q{)}^{\mathsf{T}}\mathsf{M}DX(q)$ is symmetric positive definite (The transpose $A^{\mathsf{T}}$ of a matrix, Invertible matrices and the general linear group ${\mathrm{GL}}_n(F)$), and $P$ is a generalized momentum. The image of ${\mathrm{\Phi }}_X$ is

${\mathrm{\Phi }}_X(\mathrm{\Omega }\times {\mathbb{R}}^r)=\{(X(q),\mathsf{M}DX(q)v):q\in \mathrm{\Omega },v\in {\mathbb{R}}^r\},$

the momentum chart of the presentation in the coordinates $X$; indeed $P\mapsto v=G(q{)}^{-1}P$ is a linear bijection of ${\mathbb{R}}^r$. The map ${\mathrm{\Phi }}_X$ is $C^2$: $X$ is $C^3$, so $DX$ is $C^2$; hence $G=D{X}^{\mathsf{T}}\mathsf{M}DX$ is $C^2$, and $G^{-1}$ is $C^2$ because matrix inversion is given by the adjugate divided by the nonvanishing determinant on the positive-definite matrices. The fiber coefficient $\mathsf{M}DX(q)G(q{)}^{-1}$ is $C^2$. Thus ${\mathrm{\Phi }}_X$ is a $C^2$ immersion: its derivative is injective at every point, because $DX(q)$ is injective (Generalized-coordinate maps of the model, the substituted Lagrangian, the kinetic metric and generalized forces, Submersions and immersions between Euclidean open sets) and $P\mapsto G(q{)}^{-1}P$ is a linear bijection; when $r=3N$ its derivative is invertible and it is a local $C^2$ diffeomorphism onto its open image in $W$ (equivalently $\mathsf{M}DX(q)G(q{)}^{-1}=DX(q{)}^{-\mathsf{T}}$: see the recorded identity below). We use the pointwise coordinate pullback: for a smooth $k$-form $\eta$ on $W$, set $({\mathrm{\Phi }}_X^{\ast }\eta {)}_{(q,P)}(v_1,\dots ,v_k):={\eta }_{{\mathrm{\Phi }}_X(q,P)}(D{\mathrm{\Phi }}_X(q,P)v_1,\dots ,D{\mathrm{\Phi }}_X(q,P)v_k)$, extending the formula of The pullback of a differential form to this $C^2$ map. The result has continuous coefficients. The forms of the presentation in the coordinates $X$ are

${\lambda }_X:={\mathrm{\Phi }}_X^{\ast }\lambda ,{\omega }_X:={\mathrm{\Phi }}_X^{\ast }\omega$

(The pullback of a differential form). In the coordinates $(q,P)$ on $\mathrm{\Omega }\times {\mathbb{R}}^r$ the following identities hold, and are proved in clause 2 of the symplectic-structure theorem of this page:

${\lambda }_X={\sum }_{a=1}^r{P}_{a}d{q}^a,{\omega }_X={\sum }_{a=1}^{r}d{q}^a\wedge d{P}_a;$

moreover, when $r=3N$ and $DX(q)$ is invertible, $\mathsf{M}DX(q)G(q{)}^{-1}=DX(q{)}^{-\mathsf{T}}$, so that on an open generalized-coordinate chart the chart map reads ${\mathrm{\Phi }}_X(q,P)=(X(q),DX(q{)}^{-\mathsf{T}}P)$ and the chart transition between the two descriptions of the same momentum datum is a $C^2$ local diffeomorphism.

(c) Status. The item defines the two-form datum of the declared presentation on its declared charts, together with the sign convention for fields. It asserts no closedness, no nondegeneracy, no equation of motion, no conservation law and no measured value; those are the subject of the other items of this page. The forms depend on the declared chart and its coordinate dimensions, but not on the residual forces $d$ nor on the normalization $U\mapsto U+c(t)$ of the potential representative. All objects are model objects of the idealization declared by the force branch.