Cotangent phase data, observables and action dimensions

Assume countable choice ${\mathrm{A}\mathrm{C}}_{\omega }$: every sequence of nonempty sets has a choice function (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Let $Q$ be a finite-dimensional, Hausdorff, second-countable smooth manifold without boundary, of dimension $n$. Its cotangent bundle is

$T^{\ast }Q={\coprod }_{q\in Q}{T}_q^{\ast }Q,\pi :T^{\ast }Q\to Q,\pi (q,p)=q,$

with the canonical smooth structure and induced cotangent charts supplied by the manifold construction in Tautological one-form on a cotangent bundle. The tautological one-form is ${\theta }_{(q,p)}(\xi )=p(d{\pi }_{(q,p)}\xi )$, and the canonical cotangent two-form is $\omega =-d\theta$. The exact theorem The canonical cotangent two-form is symplectic makes $(T^{\ast }Q,\omega )$ a symplectic manifold in the sense of Symplectic form and symplectic manifold. We call this pair the cotangent phase space of $Q$.

In an induced chart $(q^1,\dots ,q^n,p_1,\dots ,p_n)$, $\theta =p_{i}d{q}^i$ and $\omega =d{q}^i\wedge d{p}_i$. A mathematical observable on this phase space means a smooth function $F:T^{\ast }Q\to \mathbb{R}$; this definition does not assert that a measurement has been performed or that every such function is physically measurable. For $G\in C^{\mathrm{\infty }}(T^{\ast }Q)$, let $X_G$ be the unique vector field satisfying ${\iota }_{X_G}\omega =dG$, which exists because $\omega$ is nondegenerate, and define the canonical bracket by $\{F,G\}:=dF(X_G)$. In the induced chart, contraction gives $X_G=G_{p_i}{\mathrm{\partial }}_{q^i}-G_{q^i}{\mathrm{\partial }}_{p_i}$, so

$\{F,G\}=F_{q^i}{G}_{p_i}-F_{p_i}{G}_{q^i}\mathrm{.}$

For an interpreted particle model, assign coordinates $q^i$ their declared units and assign the conjugate covector components $p_i$ units of action divided by the units of $q^i$. Thus $\theta$ and $\omega$ have action units, conventionally joule-seconds, and the defined canonical bracket of quantities with units $[F]$ and $[G]$ has units $[F][G]\mathrm{/}(\text{action})$. These are dimensional interpretations of the model, not additional mathematical premises. The use of ${\mathrm{A}\mathrm{C}}_{\omega }$ is exactly the use in the smooth cotangent-bundle suppliers; no full Axiom of Choice is used.

Assume countable choice ACω, as in The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), for the smooth cotangent-bundle carrier convention used by this page; no full Axiom of Choice is assumed.

Let $(M,g,T)$ be a smooth, time-oriented Lorentzian four-manifold of signature $(-+++)$, with the geometric meaning fixed by Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model. Here $T$ denotes the future-cone field called $N$ in that definition. Let $I\subset \mathbb{R}$ be a nonempty open interval, let $\mathrm{\Omega }\subset {\mathbb{R}}^3$ be open, and let $U\subset M$ be open. A temporal product chart is a smooth diffeomorphism $\chi :U\to I\times \mathrm{\Omega }$. Its coordinate functions $x^0,x^1,x^2,x^3$ are the four component functions of $\chi$. Spatial indices $i,j$ range over $1,2,3$, spacetime indices $a,b$ range over $0,1,2,3$, and repeated upper/lower indices are summed. We require $x^0$ and each $x^i$ to have length units in the physical interpretation.

The chart has lapse-shift data $(N,\beta ,h)$ when $N:I\times \mathrm{\Omega }\to (0,\mathrm{\infty })$, ${\beta }^i:I\times \mathrm{\Omega }\to \mathbb{R}$, and $h_{ij}:I\times \mathrm{\Omega }\to \mathbb{R}$ are smooth, $h_{ij}=h_{ji}$, and the matrix $(h_{ij})$ is positive definite at each point, with

$g=-N^2(d{x}^0{)}^2+h_{ij}(d{x}^i+{\beta }^{i}d{x}^0)(d{x}^j+{\beta }^{j}d{x}^0)\mathrm{.}$

Write $h^{ij}$ for the inverse matrix. The normal field is

$n=N^{-1}({\mathrm{\partial }}_0-{\beta }^i{\mathrm{\partial }}_i)\mathrm{.}$

Direct contraction gives $g(n,n)=-1$; require that $n$ is future-directed in the supplied time orientation. This field of observers is distinct from the coordinate vector ${\mathrm{\partial }}_0$, which may be spacelike. The coefficients $N,{\beta }^i,h_{ij}$ are dimensionless when all four coordinates are in metres. This definition is local to the specified product chart; it asserts no global temporal foliation.

Assume countable choice ACω, as in The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), for the smooth cotangent-bundle carrier convention used by this page; no full Axiom of Choice is assumed.

Let $\mathcal{D}$ be an open subset of the time-dependent tangent bundle $\mathbb{R}\times TQ$ of a finite-dimensional smooth manifold $Q$, and let $L:\mathcal{D}\to \mathbb{R}$ be smooth. For fixed $t$, the fibre derivative is the map

$\mathbb{F}L(t,q,v)=(t,q,p),p(w)={\frac{d}{ds}\mid }_{s=0}L(t,q,v+sw),$

as in Fibre derivative or Legendre map of a Lagrangian. In induced coordinates $p_i=\mathrm{\partial }L\mathrm{/}\mathrm{\partial }{v}^i$.

The Lagrangian is regular at $(t,q,v)$ when the velocity Hessian $({\mathrm{\partial }}^{2}L\mathrm{/}\mathrm{\partial }{v}^i\mathrm{\partial }{v}^j)$ is invertible there; regularity gives only a local inverse branch. It is hyperregular on a declared domain when $\mathbb{F}L$ is a diffeomorphism from that entire domain onto an open subset of $\mathbb{R}\times T^{\ast }Q$.

It is almost regular on a declared open domain ${\mathcal{D}}_0\subseteq \mathcal{D}$, with the inherited open-submanifold structure, when the following are all assumed: $\mathbb{F}L{\mathrm{\mid }}_{{\mathcal{D}}_0}$ has constant rank; its image $N$ is an embedded submanifold of $\mathbb{R}\times T^{\ast }Q$; its corestriction ${\mathcal{D}}_0\to N$ is a surjective submersion (its differential is surjective at every point); and every fibre of that corestriction is connected. These hypotheses are not consequences of singularity alone.

Assume countable choice ACω, as in The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), for the smooth cotangent-bundle carrier convention used by this page; no full Axiom of Choice is assumed.

Fix a prescribed smooth time-oriented Lorentzian four-manifold $(M,g,T)$ with signature $(-+++)$, as in Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model. Here $T$ denotes the supplied smooth timelike field selecting the future cone, called $N$ in that geometric definition. Use the distinction among a spacetime, a worldline, a pointwise observer, an observer congruence, a tetrad and a chart given in Particle spacetime, observers and worldlines. This definition does not make $g$ dynamical.

At each $x\in M$, define $v^{\mathrm{♭}}:=g_x(v,\cdot )$ for $v\in T_{x}M$. Nondegeneracy of $g_x$ makes ${\mathrm{♭}}_x:T_{x}M\to T_x^{\ast }M$ an isomorphism; write $p^{\mathrm{♯}}:={\mathrm{♭}}_x^{-1}(p)$ for $p\in T_x^{\ast }M$, and define $g_x^{-1}(p,r):=g_x(p^{\mathrm{♯}},r^{\mathrm{♯}})$. In a chart, if $g_{ab}:=g({\mathrm{\partial }}_a,{\mathrm{\partial }}_b)$ and $(g^{ab})$ is its inverse matrix, then $p^{\mathrm{♯}}={\sum }_{a,b=0}^3{g}^{ab}{p}_b{\mathrm{\partial }}_a$. A covector $p$ is future causal here exactly when $p^{\mathrm{♯}}$ is a nonzero future-causal vector in the sense of Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model.

In this relativistic-particle model, $m>0$ is a fixed primitive mass parameter measured in kilograms, and $c>0$ is a fixed primitive conversion parameter measured in metres per second. They are parameters of the adopted physical formulation, not quantities derived by this definition. Coordinates $x^a$ in an SI chart have length units. For a cotangent vector $p=p_{a}d{x}^a$, its components $p_a$ have momentum units $\mathrm{k}\mathrm{g}\mathrm{m}{\mathrm{s}}^{-1}$ when all four coordinates are in metres. Thus $p$ as a one-form, and the canonical one-form $\theta =p_{a}d{x}^a$ on $T^{\ast }M$, have action units $\mathrm{J}\mathrm{s}$, consistently with Cotangent phase data, observables and action dimensions. Let $I\subseteq \mathbb{R}$ be an interval with nonempty interior and let $\gamma :I\to M$ be a $C^1$ curve; let $\lambda$ be its real parameter and write $V=d\gamma \mathrm{/}d\lambda$. When $V$ is future timelike, set $s:=\sqrt{-g(V,V)}$ and define the proper-time density by $d\tau =sd\lambda \mathrm{/}c$. The tangent is then nonzero. If $\tilde{\lambda }=f(\lambda )$ is an increasing $C^1$ reparametrization with $f^{\mathrm{\prime }}(\lambda )>0$, then $s_{\tilde{\lambda }}=s_{\lambda }\mathrm{/}{f}^{\mathrm{\prime }}(\lambda )$, so $s_{\tilde{\lambda }}d\tilde{\lambda }=s_{\lambda }d\lambda$. The length action $-mc\int sd\lambda$ has action units. For a future unit observer $n$ and future causal covector $p$, define the model normal-observer energy and spatial momentum covector by

$E_n(p):=-cp(n),q_n(p):=p+p(n)n^{\mathrm{♭}},$

where $n^{\mathrm{♭}}=g(n,\cdot )$. We have $q_n(p)(n)=0$; write $n^{\perp }:=\{v\in T_{x}M:g_x(v,n)=0\}$. To verify the norm convention, choose a $g_x$-orthonormal basis $(e_0,e_1,e_2,e_3)$ with $e_0$ future-directed (replace $e_0$ by $-e_0$ if necessary), and write $n=n^0{e}_0+{\sum }_{i=1}^3{n}^i{e}_i$ and $v=v^0{e}_0+{\sum }_{i=1}^3{v}^i{e}_i$. Use $n_s\cdot v_s:={\sum }_{i=1}^3{n}^i{v}^i$, $\mathrm{\mid }{n}_s{\mathrm{\mid }}^2:={\sum }_{i=1}^3(n^i{)}^2$, and $\mathrm{\mid }{v}_s{\mathrm{\mid }}^2:={\sum }_{i=1}^3(v^i{)}^2$. The future-cone convention gives $n^0>\mathrm{\mid }{n}_s\mathrm{\mid }$, and $g_x(n,n)=-1$ gives $(n^0{)}^2=1+\mathrm{\mid }{n}_s{\mathrm{\mid }}^2$. If $g_x(v,n)=0$, then $v^0=(n_s\cdot v_s)\mathrm{/}{n}^0$, so

$g_x(v,v)=\mathrm{\mid }{v}_s{\mathrm{\mid }}^2-\frac{(n_s\cdot v_s{)}^2}{(n^0{)}^2}\ge \frac{\mathrm{\mid }{v}_s{\mathrm{\mid }}^2}{(n^0{)}^2}\mathrm{.}$

Here $\mathrm{\mid }{n}_s{\mathrm{\mid }}^2\mathrm{\mid }{v}_s{\mathrm{\mid }}^2-(n_s\cdot v_s{)}^2={\sum }_{1\le i<j\le 3}(n^i{v}^j-n^j{v}^i{)}^2\ge 0$; if $v_s=0$, orthogonality also gives $v^0=0$. Thus $g_x$ is positive definite on $n^{\perp }$, and the isomorphism ${\mathrm{♯}}_x$ carries covectors annihilating $n$ onto $n^{\perp }$, so $g_x^{-1}$ is positive definite on that covector subspace. Define $\mathrm{\parallel }{q}_n(p){\mathrm{\parallel }}_n:=\sqrt{g_x^{-1}(q_n(p),q_n(p))}$. These are quantity definitions and unit conventions; they do not identify canonical momentum with mechanical momentum, assert that an apparatus measured these values, or establish a dynamical law.

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ and let $(M^{2n},\omega )$ be a smooth symplectic manifold. A finite family of smooth functions ${\chi }_1,\dots ,{\chi }_r$ defined on an open neighborhood $U\subseteq M$ is a regular primary constraint family on

$C:=\{z\in U:{\chi }_1(z)=\cdots ={\chi }_r(z)=0\}$

when $d{\chi }_1(z),\dots ,d{\chi }_r(z)$ are linearly independent at every $z\in C$. If $C$ is nonempty, the regular-level theorem A regular level set is an embedded submanifold gives $C$ its embedded-submanifold structure and $T_{z}C={\bigcap }_a\ker d{\chi }_a(z)$. If $C=\mathrm{\varnothing }$, use the unique empty smooth structure: the inclusion is an embedding and all pointwise tangent claims are vacuous; no dimension or codimension is inferred from the nonempty-level theorem. For functions $F,G$ on $U$, write $F\approx G$ when $F{\mathrm{\mid }}_C=G{\mathrm{\mid }}_C$; thus $F\approx 0$ means that $F$ vanishes on $C$, without requiring it to vanish off $C$.

Given an open interval $J\subseteq \mathbb{R}$, a smooth canonical Hamiltonian $H_c:J\times U\to \mathbb{R}$ and smooth multiplier functions $u^a:J\to \mathbb{R}$, the total generator is

$H_T(t,z):=H_c(t,z)+{\sum }_{a=1}^r{u}^a(t){\chi }_a(z)\mathrm{.}$

The functions $u^a$ are independent parameter functions; the definition does not assert that their equations determine them. If $H_c$ is an energy and ${\chi }_a$ has units $[{\chi }_a]$, then $u^a$ has units energy divided by $[{\chi }_a]$, so every term in $H_T$ has the same units. A later constraint, if generated by consistency, is not declared primary by this definition.

Let $Q$ be a finite-dimensional smooth manifold as in Cotangent phase data, observables and action dimensions, and write its cotangent phase space in induced coordinates $(q^i,p_i)$. With $\omega =d{q}^i\wedge d{p}_i$, define $X_H$ by ${\iota }_{X_H}\omega =dH$ and define the Poisson bracket by $\{F,G\}:=X_G(F)$. Then

$X_H=\frac{\mathrm{\partial }H}{\mathrm{\partial }{p}_i}\frac{\mathrm{\partial }}{\mathrm{\partial }{q}^i}-\frac{\mathrm{\partial }H}{\mathrm{\partial }{q}^i}\frac{\mathrm{\partial }}{\mathrm{\partial }{p}_i},\{F,G\}=F_{q^i}{G}_{p_i}-F_{p_i}{G}_{q^i}\mathrm{.}$

The bracket is coordinate independent and makes smooth functions a Poisson algebra. If $H_t$ is smooth on an open time-dependent domain $D\subset \mathbb{R}\times T^{\ast }Q$, each solution of $\dot{z}=X_{H_t}(z)$ has local uniqueness and smooth dependence on initial data; on its actual interval,

$\frac{d}{dt}F(t,z(t))={\mathrm{\partial }}_{t}F+\{F,H_t\}\mathrm{.}$

For parameter dependence, let $\mathrm{\Lambda }\subseteq {\mathbb{R}}^{\mathrm{\ell }}$ be open, with $\mathrm{\ell }\ge 0$, and let $H:D_{\mathrm{\Lambda }}\to \mathbb{R}$ be jointly smooth on an open $D_{\mathrm{\Lambda }}\subseteq \mathbb{R}\times T^{\ast }Q\times \mathrm{\Lambda }$. With $\mu \in \mathrm{\Lambda }$ fixed along each solution, the field is $X_{H(t,\cdot ,\mu )}$. Near any base data $(t_0,z_0,{\mu }_0)\in D_{\mathrm{\Lambda }}$, its solutions through nearby $(z,\mu )$ exist on a common compact interval about $t_0$ and depend smoothly on $(t,z,\mu )$. The unparameterized case is $\mathrm{\ell }=0$. No completeness or global flow is asserted. If $H$ is smooth, each local time-$t$ flow map preserves ω.

Let $I\subset \mathbb{R}$ be an open interval, and let $x:I\to M$ be a $C^2$ future-directed timelike worldline on a prescribed smooth, time-oriented Lorentzian background $(M,g,T)$ of signature $(-+++)$, with $m>0$ a fixed primitive mass in kg and $c>0$ in m/s. For any compact segment $[a,b]\subset I$ with $a<b$, adopt the stationary-action model on that segment

$S_{m,[a,b]}[x]=-mc{\int }_a^b\sqrt{-g(x^{\mathrm{\prime }},x^{\mathrm{\prime }})}d\lambda \mathrm{.}$

Here $I\subset \mathbb{R}$ is an open interval, $\lambda$ is its real coordinate, and $x^{\mathrm{\prime }}$ denotes $dx\mathrm{/}d\lambda$. The future-timelike condition makes $x^{\mathrm{\prime }}$ nowhere zero. On a neighborhood of $[a,b]$, $x$ is $C^2$ (so its tangent is $C^1$). If $\tilde{\lambda }=f(\lambda )$ is an increasing $C^1$ reparametrization with $f^{\mathrm{\prime }}>0$, then $\sqrt{-g(dx\mathrm{/}d\tilde{\lambda },dx\mathrm{/}d\tilde{\lambda })}d\tilde{\lambda }=\sqrt{-g(x^{\mathrm{\prime }},x^{\mathrm{\prime }})}d\lambda$ by the chain rule. An admissible variation is a family $x_{ϵ}(\lambda )$ defined for $ϵ\in (-\delta ,\delta )$, $\delta >0$, and $\lambda$ in a common neighborhood of $[a,b]$, with $x_0=x$, that is $C^2$ jointly in $(ϵ,\lambda )$, has fixed endpoint events $x_{ϵ}(a)=x(a)$ and $x_{ϵ}(b)=x(b)$, and remains future timelike for sufficiently small $\mathrm{\mid }ϵ\mathrm{\mid }$. The adopted motion principle is that on every such segment $[a,b]$, the physical worldline $x$ satisfies ${\frac{d}{dϵ}\mid }_{ϵ=0}{S}_{m,[a,b]}[x_{ϵ}]=0$ for every admissible variation through $x$. The integrand is the proper-time length density, so $S_{m,[a,b]}=-m{c}^2{\int }_a^{b}d\tau$; its dimension is action (J s). The adopted premise concerns this ideal free test body only. It is neither a theorem about all matter nor an empirical report, and supplies no backreaction or completeness claim. The flat ideal-clock specialization agrees with Adopted ideal-clock and ideal-vacuum-light interpretation; adopting this action on a curved background remains a separate premise.

The cotangent phase-space carrier convention assumes countable choice ACω only; full choice is not assumed.

On a prescribed background $(M,g,T)$, choose an open parameter interval $I$ and a compact segment $[a,b]\subset I$ with $a<b$. In a spacetime chart containing the curve and sufficiently small variations, let $x:I\to M$ be smooth near $[a,b]$, let $p(\lambda )\in T_{x(\lambda )}^{\ast }M$ be a smooth covector field along $x$, and let $e$ be smooth and positive near $[a,b]$. In this chart, write $g_{ab}=g({\mathrm{\partial }}_a,{\mathrm{\partial }}_b)$ for $a,b\in \{0,1,2,3\}$, let $(g^{ab})$ be the inverse matrix, and define $p^{\mathrm{♯}}={\sum }_{a,b=0}^3{g}^{ab}{p}_b{\mathrm{\partial }}_a$; equivalently, $g(p^{\mathrm{♯}},w)=p(w)$ for every tangent vector $w$. Adopt the distinct massless classical particle action on this segment

$S_{0,[a,b]}[x,p,e]={\int }_a^b[p_a{x}^{\mathrm{\prime }a}-e{C}_0(x,p)]d\lambda ,C_0=\frac{1}{2}{g}^{ab}{p}_a{p}_b\mathrm{.}$

Admissible variations are smooth one-parameter families of chart fields $(x_{ϵ},p_{ϵ},e_{ϵ})$ on a common neighborhood of $[a,b]$, with $x_{ϵ}(a)=x(a)$ and $x_{ϵ}(b)=x(b)$; the $p_{ϵ}$ and $e_{ϵ}$ variations are independent and have no endpoint constraints, while $e_{ϵ}>0$ for sufficiently small $\mathrm{\mid }ϵ\mathrm{\mid }$. Impose the physical branch only on stationary solutions after these variations: $p\mathrm{\ne }0$ and $p^{\mathrm{♯}}$ is future null. With all spacetime coordinates measured in metres and dimensionless $\lambda$, $p_a$ has units kg m/s, $C_0$ has units kg²m²/s² and $e$ has units s/kg, making the action J s. Other parameter units scale $e$ accordingly. This is a separate adopted model: null curves have no proper-time parameter, the model does not arise by setting $m=0$ in $P=mU$, and its positive momentum scale is additional initial data. On the nonzero branch, the vertical differential is $d_p{C}_0=p^{\mathrm{♯}}\mathrm{\ne }0$: nondegeneracy of $g^{-1}$ and $p\mathrm{\ne }0$ give $p^{\mathrm{♯}}\mathrm{\ne }0$. Hence the shell is a regular primary constraint in the sense of Regular primary constraints and total generator. This postulate is not an empirical observation or a deduction from the massive model.

The cotangent phase-space carrier convention assumes countable choice ACω only; full choice is not assumed.

Let $I\subset \mathbb{R}$ be an open interval and $(P,\omega )$ a symplectic manifold. Let ${\chi }_a:I\times P\to \mathbb{R}$ be smooth and suppose the covectors $d_z{\chi }_a(\lambda ,z)$ are linearly independent at every $z\in N_{\lambda }:=\{z:{\chi }_1(\lambda ,z)=\cdots ={\chi }_r(\lambda ,z)=0\}$; this is the regular constraint condition of Regular primary constraints and total generator on each slice. Let $H_c:I\times P\to \mathbb{R}$ be a smooth ambient extension of a specified descended Hamiltonian. Work in a Darboux chart with coordinates $(q^i,p_i)$ and local potential $\theta =p_{i}d{q}^i$, so $d\theta =-\omega$ under the convention $\omega =d{q}^i\wedge d{p}_i$; paths stay in this chart. For any compact interval $[a,b]\subset I$ with $a<b$, take paths $z:[a,b]\to P$ that extend smoothly to a neighborhood of $[a,b]$ and stay in the chart, and multipliers $u^a$ that extend smoothly to a neighborhood of $[a,b]$. Their action is

$I[z,u]={\int }_a^b[\theta (\dot{z})-H_c(\lambda ,z)-u^a(\lambda ){\chi }_a(\lambda ,z)]d\lambda \mathrm{.}$

Use $C^2$ one-parameter variations of $z$ with fixed endpoints, whose paths remain in the chart, and unrestricted $C^2$ one-parameter variations of $u$, all defined on a common neighborhood of $[a,b]$. Assume $(z,u)$ is stationary: ${\frac{d}{dϵ}\mid }_{ϵ=0}I[z_{ϵ},u_{ϵ}]=0$ for every such variation through $(z,u)$. These classes satisfy the path and fixed-endpoint hypotheses of Stationarity of the time-dependent action is equivalent to the vanishing of the Euler-Lagrange expression. Stationarity implies $z(\lambda )\in N_{\lambda }$ and $\dot{z}=X_{H_c(\lambda ,\cdot )}+u^a(\lambda )X_{{\chi }_a(\lambda ,\cdot )}$. Along this trajectory, preservation of the moving level set requires

${\mathrm{\partial }}_{\lambda }{\chi }_a+\{{\chi }_a,H_c(\lambda ,\cdot )\}+u^b(\lambda )\{{\chi }_a,{\chi }_b\}=0\text{at }(\lambda ,z(\lambda ))\mathrm{.}$

For constraints independent of $\lambda$, the first term is zero.

Let $(P^{2n},\omega )$ be symplectic, and let $N=\{{\chi }_1=\cdots ={\chi }_r=0\}$ be a regular embedded level surface with $\{{\chi }_a,{\chi }_b\}=0$ on $N$. Then $\ker (\omega {\mathrm{\mid }}_N)$ is exactly the rank-$r$ distribution spanned by the restricted Hamiltonian vector fields $X_{{\chi }_a}$. This distribution is involutive. Around every point of $N$, its leaves admit smooth Frobenius coordinates $(s^1,\dots ,s^r,y^1,\dots ,y^{2n-2r})$, and the local leaf space has a unique smooth closed nondegenerate two-form ${\omega }_{\mathrm{r}\mathrm{e}\mathrm{d}}$ whose pullback is $\omega {\mathrm{\mid }}_N$. Thus regular first-class constraints yield a local symplectic quotient. No global orbit-space regularity follows from these hypotheses.

Let $L(q,v)$ be $C^2$ on a conic open subset of a tangent bundle, and suppose $L(q,av)=aL(q,v)$ for every $a>0$ in its domain. At each nonzero $v$,

$v^i{L}_{v^i}=L,L_{v^i{v}^j}{v}^i=0,E_L:=v^i{L}_{v^i}-L=0.$

Consequently the velocity Hessian is singular and the fibre derivative cannot be a local diffeomorphism there; a regular Legendre equivalence theorem cannot be applied. These conclusions do not imply that the motion is static or that the Lagrangian has a specified constraint surface.

Let $L(t,q,v)$ be smooth on an open velocity domain and let $\mathbb{F}L(t,q,v)=(t,q,L_v)$ be almost regular in the sense that it is a surjective submersion onto a specified embedded image $N$ and every fibre is connected. Define $E_L=v^i{L}_{v^i}-L$. Then there is a unique smooth function $h:N\to \mathbb{R}$ with $E_L=h\circ \mathbb{F}L$. The assertion is local in the stated domains; no global inverse velocity map is asserted.

Let $(P,\omega )$ be symplectic and let ${\chi }_1,\dots ,{\chi }_{2k}$ be smooth on an open set $U$, with $C_{ab}=\{{\chi }_a,{\chi }_b\}$ invertible at every point of $U$. Write $C^{ab}$ for its inverse and $N=\{\chi =0\}\cap U$. Then $N$ is an embedded symplectic submanifold, and the bracket

$\{F,G{\}}_D=\{F,G\}-\{F,{\chi }_a\}{C}^{ab}\{{\chi }_b,G\}$

is the Poisson bracket of the restricted symplectic form on $N$. It is independent of ambient extensions of $F,G$, and every ${\chi }_a$ is a Casimir. This is local and assumes invertibility on $U$; it does not cover singular points.