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Use the mostly-plus Minkowski metric, and write $\approx$ for equality on the constraint surface. Assume that the mechanical constraints are locally independent. They are first-class constraints whenThus their Poisson brackets vanish on the constraint surface, and their Hamiltonian flows preserve that surface. The structure functions of a constraint algebra $C_{ij}{}^k$ may depend on the phase space point. The finite real span of the constraints is a Lie algebra if it closes with constant structure coefficients, in a suitable choice of generators. The Jacobi identity then gives the usual conditions on the structure constants of a Lie algebra. With general structure functions the finite real span need not close, even though the Poisson bracket of all smooth functions is itself a Lie bracket.

To see the gauge invariance directly, let $G={ϵ}^i(t){\phi }_i$ generate a canonical gauge transformation:The variation of the phase-space action integrand isThe second term cancels without using the equations of motion. Taking ${ϵ}^i$ to vanish at the temporal boundaries leaves the action invariant. Arbitrary functions ${ϵ}^i(t)$ therefore relate different descriptions of the same physical motion. This reasoning also works with structure functions; constant structure coefficients are only needed for the finite-dimensional Lie algebra claim.

For a closed string, choose $0\le \sigma <2\pi$ and periodic fields. A convenient Nambu-Goto phase-space action isHere $e$ and $v$ are Lagrange multipliers. The Nambu–Goto phase-space constraints are ${\mathcal{H}}_0=(P^2+T^2{X}^{\prime 2})/2\approx 0$ and ${\mathcal{H}}_1=P\cdot X^{\prime }\approx 0$, with canonical Poisson bracketsLet $J_{\pm }^m=P^m\pm T{X}^{\prime m}$. Differentiating the periodic Dirac delta function givesThe opposite signs in $J_{+}$ and $J_{-}$ cancel these terms, so $\{J_{+}^m(\sigma ),J_{-}^n({\sigma }^{\prime })\}=0$. Replace the original constraints by the equivalent chiral densitiesTheir mixed Poisson brackets vanish. Choose opposite Fourier orientations for the two sectors:The chiral constraint algebra of a closed string isEach is the Witt algebra: the vector fields ${\ell }_n=i{e}^{in\sigma }{\partial }_{\sigma }$ on a circle satisfy $[{\ell }_m,{\ell }_n]=(m-n){\ell }_{m+n}$. Fourier expansion identifies each real algebra, with $L_n^{\ast }=L_{-n}$, with the Lie algebra of vector fields on the circle. The two commuting copies give ${\mathrm{Diff}}_1\oplus {\mathrm{Diff}}_1$, not a quantum central extension.

For an open string, allowed boundary conditions must remove the endpoint term in the variation of the action, consistently with the allowed endpoint variations. The spatial boundary term isIt expresses the open-string endpoint momentum flux. In the temporal gauge for a string $X^0=t$, take a boundary-adapted parametrization with $v=0$ at the ends. Fixing $X^i(t,a)=0$ gives $\delta X^i(t,a)=0$, a Dirichlet boundary condition. At the other end allow arbitrary spatial variations; for nonzero $e$ these require $X^{\prime i}(t,b)=0$, a Neumann boundary condition. Also $X^{\prime 0}=0$ in this temporal gauge for a string, so $X^{\prime m}(t,b)=0$. The constraint at this free-end string boundary condition reduces to $P^2=0$. Hamilton's equation $\dot{X}=eP+v{X}^{\prime }$ consequently gives ${\dot{X}}^2=0$ there. Since ${\dot{X}}^0=1$, the free endpoint has spatial speed one. This is the null motion of a free string endpoint.

A straight rotating string with one fixed endpoint supplies the required solution in at least two spatial dimensions. Set $e=1/T$, $v=0$, andTake $P_m=T{\dot{X}}_m$. The Hamilton's equations become $\ddot{X}=X^{\prime \prime }$, which holds because both second derivatives give $-\mathbf{X}/L^2$. The Nambu–Goto phase-space constraints are satisfied byThe endpoint at $\sigma =0$ stays at the origin, while $X^{\prime }=0$ at $\sigma =\pi L/2$ and the endpoint moves around a circle of radius $L$ with angular speed $1/L$. At each time the whole string lies on a straight radial segment. Its spatial proper length isThe velocity is everywhere perpendicular to the segment, so this also equals the sum of local rest-frame lengths. The induced worldsheet metric becomes degenerate at the null free endpoint, as expected for the limiting free-end solution.

Use Minkowski spacetime signature $(-,+,+,+)$ and a closed spatial parameter of period $2\pi$. Choose the future-directed branch with positive lapse $e$.

Canonical dynamics and gauge freedom. Variation of momentum and embedding in the Nambu-Goto phase-space action givesThe Lagrange multipliers impose the Nambu–Goto phase-space constraints $P^2+T^2{X}^{\prime 2}=0$ and $P\cdot X^{\prime }=0$. These two first-class constraints reflect freedom to relabel time and space on the same string worldsheet. The canonical Hamiltonian is a linear combination of constraints, with arbitrary multiplier functions. These functions specify a coordinate description rather than additional propagating fields. No explicit gauge transformation or Poisson-bracket calculation is required.

Eliminating auxiliary variables. Let $V=\dot{X}-u{X}^{\prime }$. Eliminating momentum using $P=V/e$ leavesOn a patch with spacelike spatial tangent, variation of $u$ gives $V\cdot X^{\prime }=0$, hence $u=(\dot{X}\cdot X^{\prime })/X^{\prime 2}$. For the induced worldsheet metric $g_{ab}={\partial }_{a}X\cdot {\partial }_{b}X$, this gives $V^2=\det g/X^{\prime 2}$. Variation of $e$ then gives $e=\sqrt{-\det g}/(T{X}^{\prime 2})$ on the positive branch. Substitution yieldsThis is minus string tension times Lorentzian worldsheet area. Vary the area using $\delta \sqrt{-g}=\frac{1}{2}\sqrt{-g}{g}^{ab}\delta g_{ab}$ and integrate by parts. The Nambu–Goto equations of motion areThe auxiliary-variable elimination and this metric form apply on nondegenerate timelike patches.

Circular motion, length and energy. For the circular embedding, direct differentiation givesAway from collapse, $\sqrt{-g}{g}^{ab}=\mathrm{diag}(-1,1)$, so the equations become ${\ddot{X}}^m-X^{\prime \prime m}=0$. The time and out-of-plane coordinates satisfy them immediately. For $Z=X^1+i{X}^2$, both $\ddot{Z}$ and $Z^{\prime \prime }$ are $-Z$. The relations $\dot{X}\cdot X^{\prime }=0$ and ${\dot{X}}^2+X^{\prime 2}=0$ verify the Virasoro constraints. Equivalently, $e=1/T$, $u=0$, $P=T\dot{X}$ solve the phase-space equations and constraints at all times.

The ring stays in a fixed plane, with radius $R|\cos t|$. It contracts to a point and re-expands. Each labeled point moves radially with speed $|\sin t|$ in target time $X^0=Rt$. The geometric ring repeats after target-time interval $\pi R$, although the labels have then shifted by half a circumference. It is not rigidly rotating. Velocity is perpendicular to the tangent, so simultaneous spatial arclength is also the local proper length along the string. Its length and conserved energy areAway from collapse, the same energy follows from $T\int dl/\sqrt{1-v^2}$. Increasing kinetic energy compensates the shrinking length. At collapse the induced metric degenerates; the area-form equation alone is undefined there. The regular phase-space solution supplies the continuation and the limiting constant energy. This is a pulsating circular string.

Endpoint variation. For an open string, integration by parts produces the boundary termIts coefficient is the open-string endpoint momentum flux. Evaluate this ungauge-fixed variation in a boundary-adapted temporal gauge for a string $X^0=t$. Then $X^{\prime 0}=0$ and the time equation gives $P^0=1/e\ne 0$. The free variation of time forces $u{P}_0=0$, hence $u=0$ at each end. Since $e$ is nonzero, the remaining spatial term requiresFree variation in every spatial direction gives ${\vec{X}}^{\prime }=0$, and together with $X^{\prime 0}=0$ gives the free-end string boundary condition $X^{\prime m}=0$. The general starting condition is the flux condition; the shift term should not be silently discarded before choosing the gauge.

Target-space charges. The Noether charges for translations and Lorentz transformations are the target-space Noether charges of a stringWriting $F_m=e{T}^2{X}_m^{\prime }+u{P}_m$, the canonical equations implyFor the second identity, antisymmetry cancels the $e{P}_m{P}_n$ terms. Integration by parts then cancels the $u{X}^{\prime }P$ terms and the symmetric $e{T}^2{X}_m^{\prime }{X}_n^{\prime }$ terms. For free ends $F_m=0$ individually, so both charges are constant. In conformal gauge, the same result follows from $u=0$ and $X^{\prime m}=0$. Target-space energy is $-{\mathcal{P}}_0$.

Other possibilities fix selected spatial directions by Dirichlet boundary conditions, while leaving Neumann boundary conditions in the permitted tangent directions. These mixed conditions describe endpoints on a D-brane; the two ends can lie on different branes. Time stays free under the given assumption. Fixed supports may absorb momentum in their Dirichlet directions and break corresponding translations or Lorentz symmetries. More general force-law conditions require an additional boundary interaction.

Define the open-string endpoint momentum flux along the open string by $J_m=e{T}^2{X}_m^{\prime }+u{P}_m$. Variation of the Nambu-Goto phase-space action givesThe two last equations are the Virasoro constraints. Integration of the momentum equation givesThus the necessary and sufficient condition for total momentum conservation is equality of the two endpoint fluxes, component by component. Vanishing of both fluxes is a sufficient local boundary condition.

The spatial boundary term in the action variation is $-\int dt[J_m\delta X^m{]}_0^{\pi }$. For free ends the endpoint variations are arbitrary, soIn a gauge with $u=0$ and nonzero $e$ this reduces to the usual Neumann boundary condition $X^{\prime m}=0$. The conclusion is conservation of total momentum, not that total momentum must vanish. The PDF contains the derivative in this conclusion; the TeX transcription drops it.

For endpoints confined to $x^1=0$, the Dirichlet boundary condition fixes $\delta X^1=0$. It therefore imposes no requirement that $J_1$ vanish. Preserving the fixed position requires ${\dot{X}}^1=e{P}^1+u{X}^{\prime 1}=0$ at each endpoint, but that is a different condition. For example, in $u=0$ gauge the endpoint momentum $P^1$ is zero while $e{T}^2{X}^{\prime 1}$ can be nonzero. Consequently ${\mathcal{P}}_1$ is generally not conserved: the normal endpoint forces transfer momentum to the hyperplane. A dynamical D-brane recoils and carries the missing momentum. If the hyperplane is idealized as fixed, its external support absorbs the momentum. The combined string-and-support system conserves momentum; freely varying directions tangent to the hyperplane retain their vanishing-flux condition.