Symplectic geometry and canonical methods

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.

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, charts and forms 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: masses $m_1,\dots ,m_N>0$, open configuration domain $Q\subseteq {\mathbb{R}}^{3N}$ and mass operator $\mathsf{M}$ (Configuration domain of $N$ labeled particles: open admissible sets, regular holonomic constraints and the mass operator), time window $I_0$, $C^2$ potential $U$ 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 and SI unit conventions of the non-relativistic classical mechanics framework), the Cartesian phase domain $W=Q\times {\mathbb{R}}^{3N}$ with $n=3N$, its canonical one-form $\lambda$ and two-form $\omega =-d\lambda$, and for each generalized-coordinate map $X:\mathrm{\Omega }\to Q$ with $r$ coordinates and kinetic metric $G$ the chart map ${\mathrm{\Phi }}_X(q,P)=(X(q),\mathsf{M}DX(q)G(q{)}^{-1}P)$ with pullbacks ${\lambda }_X={\mathrm{\Phi }}_X^{\ast }\lambda$ and ${\omega }_X={\mathrm{\Phi }}_X^{\ast }\omega$ (Generalized-coordinate maps of the model, the substituted Lagrangian, the kinetic metric and generalized forces, The momentum of a potential presentation: the fibre derivative of the mechanical Lagrangian, the Cartesian momentum Mv and generalized momenta, Hyperregular potential presentations and their inverse Legendre branches in the model's charts). Then:

1. (Cartesian chart) $\omega$ is the standard two-form of the phase domain $W$ in the sense of A phase domain, its canonical one-form and its standard symplectic two-form: $d\omega =0$ and $\omega$ is nondegenerate at every point of $W$; hence $(W,\omega )$ is a symplectic manifold in the sense of Symplectic form and symplectic manifold.
2. (generalized-coordinate chart) On $\mathrm{\Omega }\times {\mathbb{R}}^r$,

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

consequently ${\omega }_X$ is closed and nondegenerate at every point, so ${\mathrm{\Phi }}_X^{\ast }\omega$ is the standard two-form of the phase domain $\mathrm{\Omega }\times {\mathbb{R}}^r$ and ${\mathrm{\Phi }}_X$ preserves the two-form. 3. (full-dimensional chart transitions) Let $X_1,X_2$ be generalized-coordinate charts with $r=3N$, and restrict their momentum chart maps to neighborhoods on which each ${\mathrm{\Phi }}_{X_j}$ is a diffeomorphism onto an open subset of the Cartesian phase domain. On every overlap of those image neighborhoods, the genuine phase-chart transition $\mathrm{\Theta }={\mathrm{\Phi }}_{X_2}^{-1}\circ {\mathrm{\Phi }}_{X_1}$ preserves the two-form. The difference ${\lambda }_{X_1}-{\mathrm{\Theta }}^{\ast }{\lambda }_{X_2}$ is a closed one-form there; no exactness is asserted. Lower-dimensional generalized-coordinate immersions have the symplectic pullback form of clause 2, but are not transitions between open full-dimensional phase charts.

The statement is a conditional deduction from the adopted force branch, the declared presentation and the cited mathematics. It asserts no equation of motion, no conservation law, no completeness of any flow and no empirical content.

• What is and is not proved here. Clauses 1-3 are chart calculus on the declared presentation. No statement is made about the existence of a global diffeomorphism between the charts (${\mathrm{\Phi }}_X$ need not be injective for a constraint parametrization), about dynamics, or about the completeness of any flow; the field-theoretic consequences are the subject of Cartesian model motions on the potential branch and declared generalized-coordinate Lagrange equations integrate their Hamiltonian fields; the forced Cartesian field is Hamiltonian exactly when the residual force has a velocity-independent potential and the subsequent items of this page.

• Cotangent chart transitions and general gauge differences. For the specific transitions of clause 3, both one-forms are pullbacks of the same Cartesian $\lambda$, so step 4.1 gives ${\lambda }_{X_1}-{\mathrm{\Theta }}^{\ast }{\lambda }_{X_2}=0$. For a general canonical transformation preserving only $\omega$, the corresponding difference of canonical one-forms is closed and can be nonexact on a multiply connected domain; the cylinder obstruction in Scope of the symplectic and canonical methods for a potential presentation: locality of generating functions, projections and determinant tests, absent global claims, and empirical status concerns this more general situation.

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, the phase domains and the 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 Cartesian phase domain $W=Q\times {\mathbb{R}}^{3N}$ with $n=3N$, its two-form $\omega$, and for each generalized-coordinate map $X:\mathrm{\Omega }\to Q$ with $r$ coordinates the chart map ${\mathrm{\Phi }}_X$ and the form ${\omega }_X$ (Generalized-coordinate maps of the model, the substituted Lagrangian, the kinetic metric and generalized forces, The momentum of a potential presentation: the fibre derivative of the mechanical Lagrangian, the Cartesian momentum Mv and generalized momenta, Configuration domain of $N$ labeled particles: open admissible sets, regular holonomic constraints and the mass operator). All differentiability classes are those of $C^k$ Euclidean maps and diffeomorphisms.

(a) Canonical coordinates. Let $W_0\subseteq W$ (or $W_0\subseteq \mathrm{\Omega }\times {\mathbb{R}}^r$) be nonempty open, and let

$\theta =({\theta }^1,\dots ,{\theta }^{2m}):W_0⟶V\subseteq {\mathbb{R}}^{2m}$

be a $C^1$ chart of $W_0$ onto an open set $V$ (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), with $m$ the half-dimension of $W_0$. Write $\sigma =\omega {\mathrm{\mid }}_{W_0}$ and $\alpha =\lambda {\mathrm{\mid }}_{W_0}$ in the Cartesian case, and $\sigma ={\omega }_X{\mathrm{\mid }}_{W_0}$ and $\alpha ={\lambda }_X{\mathrm{\mid }}_{W_0}$ in the generalized-coordinate case. Each coordinate ${\theta }^a$ has a declared dimension ${\eta }_a$ and unit; its paired coordinate ${\theta }^{m+a}$ has dimension $\mu {\mathrm{\ell }}^2{\tau }^{-1}{\eta }_a^{-1}$ and a declared compatible unit, so every summand below has action dimension. Coordinate formulas use the numerical representatives in these fixed units. The coordinates $\theta$ are canonical coordinates for the presentation's two-form on $W_0$ when the two-form takes the canonical form

$\sigma ={\sum }_{a=1}^{m}d{\theta }^a\wedge d{\theta }^{m+a}$

pointwise on $W_0$, using the finite-coordinate form and pullback conventions of Finite-regularity differential forms, Hamiltonian fields and canonical maps in phase charts; equivalently, when the pullback identity ${\theta }^{\ast }{\omega }_V^{\mathrm{s}\mathrm{t}}=\sigma$ holds for the standard two-form ${\omega }_V^{\mathrm{s}\mathrm{t}}$ of $V$ (A phase domain, its canonical one-form and its standard symplectic two-form, Finite-regularity differential forms, Hamiltonian fields and canonical maps in phase charts), equivalently when $D\theta (z{)}^{\mathsf{T}}JD\theta (z)=J$ for the standard symplectic matrix $J$ at every $z\in W_0$ (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 $X_H$). A canonical chart of the presentation is a chart whose coordinates are canonical in this sense; the equivalence with the canonical maps defined later on this page is recorded with those definitions.

(b) The model's own charts are canonical. The Cartesian coordinates $(x,p)$ on $W$ are canonical coordinates: by clause 1 of The two-form of a potential presentation is closed and nondegenerate, and each generalized-coordinate momentum chart carries the same symplectic structure: ${\mathrm{\Phi }}_X^{\ast }\lambda =\sum P_{a}d{q}^a$ and ${\mathrm{\Phi }}_X^{\ast }\omega =\sum d{q}^a\wedge d{P}_a$ the two-form is $\omega ={\sum }_{a}d{x}^a\wedge d{p}_a$. For a generalized-coordinate map, the coordinates $(q,P)$ on $\mathrm{\Omega }\times {\mathbb{R}}^r$ are canonical coordinates for the pulled-back form ${\omega }_X$: by clause 2 of The two-form of a potential presentation is closed and nondegenerate, and each generalized-coordinate momentum chart carries the same symplectic structure: ${\mathrm{\Phi }}_X^{\ast }\lambda =\sum P_{a}d{q}^a$ and ${\mathrm{\Phi }}_X^{\ast }\omega =\sum d{q}^a\wedge d{P}_a$ one has ${\omega }_X={\sum }_{a}d{q}^a\wedge d{P}_a$, and when the map is a genuine chart ($r=3N$) the chart transition is a change of canonical coordinates on overlaps.

(c) Canonical one-form up to gauge at classical differentiability. For a $C^2$ canonical chart $\theta$, the source canonical one-form $\alpha$ specified in clause (a) from 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 is said to be in canonical form up to gauge in the canonical coordinates $\theta$ when the difference $\alpha -{\sum }_a{\theta }^{m+a}d{\theta }^a$ is a closed one-form on $W_0$; exactness of that difference is not required. For the presentation's own charts the difference of the two canonical one-forms is closed by clause 3 of The two-form of a potential presentation is closed and nondegenerate, and each generalized-coordinate momentum chart carries the same symplectic structure: ${\mathrm{\Phi }}_X^{\ast }\lambda =\sum P_{a}d{q}^a$ and ${\mathrm{\Phi }}_X^{\ast }\omega =\sum d{q}^a\wedge d{P}_a$. These charts are $C^2$ or smoother. For a merely $C^1$ chart clause (a) retains its full pointwise meaning, but the coefficients of the pulled-back one-form may be only continuous, so this clause does not apply the classical exterior derivative to them.

(d) Status and warnings. The item defines a coordinate condition on a declared chart and records the one fact that the presentation's own momentum charts satisfy it. It asserts no existence of canonical coordinates beyond the model's charts, no global statement on multiply connected phase domains, no dynamics and no measured value. A chart that is merely a diffeomorphism need not be canonical; the determinant test $\mathrm{\mid }\det D\theta \mathrm{\mid }=1$ is necessary but, in dimension $2m\ge 4$, not sufficient for canonicity (the scope remark of this page records the witness).

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 and 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: the Cartesian phase domain $W=Q\times {\mathbb{R}}^{3N}$, its two-form $\omega$, a generalized-coordinate chart map ${\mathrm{\Phi }}_X$ and the form ${\omega }_X$ (Generalized-coordinate maps of the model, the substituted Lagrangian, the kinetic metric and generalized forces); the presentation is hyperregular on every chart with the Hamiltonian $H$ of The Hamiltonian of a hyperregular potential presentation: the Legendre transform of the mechanical Lagrangian, its domain and its dimensions and Every potential presentation is hyperregular in its charts, with inverse branch $v={\mathsf{M}}^{-1}p$ (or $G(q{)}^{-1}p$) and the explicit kinetic-plus-potential Hamiltonian,

$H(t,x,p)=\frac{1}{2}p\cdot {\mathsf{M}}^{-1}p+U(t,x),\tilde{H}(t,q,P)=\frac{1}{2}P\cdot G(q{)}^{-1}P+\tilde{U}(t,q)\mathrm{.}$

Differentiability classes are those of $C^k$ Euclidean maps and diffeomorphisms, and gradients those of The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case.

(a) The field of a phase function. Let $W_0\subseteq W$ be nonempty open and let $f:W_0\to \mathbb{R}$ be of class $C^2$. By clause 1 of The two-form of a potential presentation is closed and nondegenerate, and each generalized-coordinate momentum chart carries the same symplectic structure: ${\mathrm{\Phi }}_X^{\ast }\lambda =\sum P_{a}d{q}^a$ and ${\mathrm{\Phi }}_X^{\ast }\omega =\sum d{q}^a\wedge d{P}_a$ the form $\omega$ is nondegenerate on $W_0$, so there is exactly one $C^1$ vector field $X_f$ on $W_0$ with

${\iota }_{X_f}\omega =df$

(the sign convention of Hamiltonian vector field and Hamiltonian function and The Hamiltonian field of a $C^2$ function on a phase domain is $J\mathrm{\nabla }H$, and the Poisson bracket is ${\sum }_i({\mathrm{\partial }}_{q^i}f{\mathrm{\partial }}_{p_i}g-{\mathrm{\partial }}_{p_i}f{\mathrm{\partial }}_{q^i}g)$); it is the Hamiltonian field of $f$ for the presentation. In the Cartesian coordinates $(x,p)$, writing $\mathrm{\nabla }f=(f_{x^1},\dots ,f_{x^{3N}},f_{p_1},\dots ,f_{p_{3N}})$,

$X_f=(f_p,-f_x)=J\mathrm{\nabla }f,J=\left(\begin{array}{cc}{0}_{3N}& I_{3N}\\ -I_{3N}& 0_{3N}\end{array}\right);$

in any canonical chart $({\theta }^1,\dots ,{\theta }^{2m})$ of the presentation the same field reads $X_f=(f_{{\theta }^{m+1}},\dots ,f_{{\theta }^{2m}},-f_{{\theta }^1},\dots ,-f_{{\theta }^m})$. A vector field on $W_0$ is called Hamiltonian for the presentation when it is $X_f$ for some $C^2$ function $f$; in particular $X_f=0$ exactly when $f$ is locally constant on $W_0$. Vector fields are read as tangent-bundle sections of A smooth vector field is a smooth section of the tangent bundle.

(b) The field of the branch Hamiltonian. For each $t\in I_0$ the function $H_t:=H(t,\cdot )$ is $C^2$ on $W$, and the Hamiltonian field of the presentation at time $t$ is $X_{H_t}$:

$X_{H_t}(x,p)=({\mathsf{M}}^{-1}p,-{\mathrm{\nabla }}_{x}U(t,x)),$

whose components are a velocity and a force. In the coordinates $X$ the corresponding field is $X_{{\tilde{H}}_t}(q,P)=(G(q{)}^{-1}P,-{\mathrm{\partial }}_q\tilde{H}(t,q,P))$, where the second slot retains the kinetic contribution $-\frac{1}{2}P\cdot {\mathrm{\partial }}_q(G(q{)}^{-1})P$ when the metric varies. If $d\equiv ̸0$ the model's forced evolution carries the additional momentum source $d$; the corresponding vector field $(x,p)\mapsto ({\mathsf{M}}^{-1}p,-{\mathrm{\nabla }}_{x}U(t,x)+d(t,x,{\mathsf{M}}^{-1}p))$ is generally not a Hamiltonian field for the presentation, and it is one exactly when the residual force has a potential independent of the velocity slot (the criterion is proved in clause 4 of the corresponding theorem of this page).

(c) Time dependence and status. For a time-dependent $H$ the assignment $t\mapsto X_{H_t}$ is a time-dependent field on the phase domain; each $X_{H_t}$ is $C^1$, hence locally Lipschitz in the state variable, and no completeness of its integral curves is asserted here. The item defines the field datum and its chart expressions; the identification of the model's motions with the integral curves, and all conservation or invariance properties, are the subject of the other items of this page. The field is defined by the intrinsic equation ${\iota }_X\omega =dH$ and therefore does not depend on the canonical chart used to compute it.

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, charts and Hamiltonian field 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 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: masses $m_1,\dots ,m_N>0$ and open configuration domain $Q\subseteq {\mathbb{R}}^{3N}$ with mass operator $\mathsf{M}$ (Configuration domain of $N$ labeled particles: open admissible sets, regular holonomic constraints and the mass operator), time window $I_0$, $C^2$ potential $U$ 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 and SI unit conventions of the non-relativistic classical mechanics framework), the Cartesian Hamiltonian $H(t,x,p)=\frac{1}{2}p\cdot {\mathsf{M}}^{-1}p+U(t,x)$ and, for a generalized-coordinate map $X:\mathrm{\Omega }\to Q$ with kinetic metric $G=D{X}^{\mathsf{T}}\mathsf{M}DX$, the chart Hamiltonian $\tilde{H}(t,q,P)=\frac{1}{2}P\cdot G(q{)}^{-1}P+\tilde{U}(t,q)$ (The Hamiltonian of a hyperregular potential presentation: the Legendre transform of the mechanical Lagrangian, its domain and its dimensions, Every potential presentation is hyperregular in its charts, with inverse branch $v={\mathsf{M}}^{-1}p$ (or $G(q{)}^{-1}p$) and the explicit kinetic-plus-potential Hamiltonian, Generalized-coordinate maps of the model, the substituted Lagrangian, the kinetic metric and generalized forces, The momentum of a potential presentation: the fibre derivative of the mechanical Lagrangian, the Cartesian momentum Mv and generalized momenta). Let $J\subseteq I_0$ be a nonempty open interval. For clause 4, a time-dependent phase function is a map $f:I_0\times W\to \mathbb{R}$ of class $C^2$; write $f_t(z):=f(t,z)$ and use 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 to define the associated family of Hamiltonian fields $t\mapsto X_{f_t}$. The forced field is called Hamiltonian on this full chart when it equals $X_{f_t}$ for every $t\in I_0$. Then:

1. (potential branch, Cartesian chart) Assume $d\equiv 0$ on the state domain. For a $C^2$ curve $x:J\to Q$ with $p(t):=\mathsf{M}{x}^{\mathrm{\prime }}(t)$ the following are equivalent: (a) $x$ is a motion of the model (The associated first-order system, motions, maximal motions and the Newtonian flow of a force model); (b) $\dot{x}(t)=H_p(t,x(t),p(t))$ and $\dot{p}(t)=-H_x(t,x(t),p(t))$ for every $t\in J$; (c) the curve $t\mapsto (x(t),p(t))$ is an integral curve of the time-dependent field $t\mapsto X_{H_t}$, i.e. satisfies $\dot{z}(t)=X_{H_t}(z(t))$ with $z=(x,p)$ (First-order systems, initial value problems, and solutions on intervals).
2. (generalized-coordinate chart) Let $X:\mathrm{\Omega }\to Q$ be a generalized-coordinate map and let $q:J\to \mathrm{\Omega }$ be a $C^2$ curve with $x:=X\circ q$ and $P(t):=G(q(t))q^{\mathrm{\prime }}(t)$. Then the declared Lagrange equations with residual generalized force, $\mathrm{\Lambda }(\tilde{L})(q)(t)=Q^{\mathrm{r}\mathrm{e}\mathrm{s}}(t,q(t),q^{\mathrm{\prime }}(t))$, are equivalent to $q^{\mathrm{\prime }}(t)={\tilde{H}}_P(t,q(t),P(t))$ and $P^{\mathrm{\prime }}(t)=-{\tilde{H}}_q(t,q(t),P(t))+Q^{\mathrm{r}\mathrm{e}\mathrm{s}}(t,q(t),q^{\mathrm{\prime }}(t))$; in particular, when $d\equiv 0$, they are equivalent to $(q,P)$ being an integral curve of $X_{{\tilde{H}}_t}$.
3. (residual forces, Cartesian chart) For a $C^2$ curve $x:J\to Q$ with $p=\mathsf{M}{x}^{\mathrm{\prime }}$, the model's forced equation $\mathsf{M}{x}^{\mathrm{\prime }\mathrm{\prime }}=-{\mathrm{\nabla }}_{x}U(t,x)+d(t,x,x^{\mathrm{\prime }})$ holds on $J$ if and only if $\dot{x}=H_p$ and $\dot{p}=-H_x+d(t,x,{\mathsf{M}}^{-1}p)$ on $J$.
4. (when is the forced field Hamiltonian?) The forced field $(x,p)\mapsto ({\mathsf{M}}^{-1}p,-{\mathrm{\nabla }}_{x}U(t,x)+d(t,x,{\mathsf{M}}^{-1}p))$ is Hamiltonian in the preceding time-dependent sense if and only if there is a $C^2$ function $V:I_0\times Q\to \mathbb{R}$ with $d(t,x,v)=-{\mathrm{\nabla }}_{x}V(t,x)$ for every $(t,x,v)$ of the state domain; in that case the phase function is $H+V$ up to an independently chosen additive $C^2$ function of $t$ on each connected component of $Q$, and the model's forced motions are the motions of the potential presentation $(U+V,0)$.