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With string tension $T=1/(2\pi {\alpha }^{\prime })$, the free-ended open string $X^0=a\tau$, $X^1+i{X}^2=a\cos \sigma e^{i\tau }$, $0\le \sigma \le \pi$, is straight at every target time. Its Nambu–Goto equations of motion and Virasoro constraints hold, and its ends move at light speed. Integrating the target-space Noether charges of a string gives $M=\pi Ta$ and $J=\pi T{a}^2/2$. Thus $J={\alpha }^{\prime }{M}^2$. Extending the same map to $0\le \sigma \le 2\pi$ describes a double-covered folded closed string, with twice the mass and twice the angular momentum, hence $J={\alpha }^{\prime }{M}^2/2$. The different slopes record the different boundary topology.

Use Minkowski spacetime with signature $(-,+,\dots ,+)$, and write the string tension as $T=1/(2\pi {\alpha }^{\prime })$. If $X^{\mu }(\tau ,\sigma )$ is the string embedding map, its induced worldsheet metric is ${\gamma }_{ab}={\partial }_{a}X\cdot {\partial }_{b}X$. The Nambu–Goto action isThe first variation uses $\delta {\gamma }_{ab}=2{\partial }_{(a}{X}_{\mu }{\partial }_{b)}\delta X^{\mu }$ and therefore givesAn integration by parts yields the bulk Nambu–Goto equations of motion and the endpoint term:For a closed string, periodicity cancels the endpoint term. For a freely moving open string, arbitrary endpoint variations require zero momentum flux, which is the free-end string boundary condition.

Choose orthogonal conformal coordinates on the nondegenerate part of the worldsheet, so that ${\gamma }_{\tau \sigma }=0$ and ${\gamma }_{\tau \tau }+{\gamma }_{\sigma \sigma }=0$. The Nambu–Goto equations of motion then become a wave equation, with the accompanying Virasoro constraints:Here a dot and a prime mean ${\partial }_{\tau }$ and ${\partial }_{\sigma }$. The last two equations are essential: a solution of the wave equation alone need not be a relativistic fundamental string motion.

For a free-ended open string, take $0\le \sigma \le \pi$ and $a>0$, and setwith the other coordinates constant. Every component satisfies the wave equation, andMoreover $X^{\prime }=0$ at both endpoints, giving the Neumann boundary condition. At fixed target time $t=a\tau$, the spatial image is the straight segmentThus its angular velocity is $1/a$. The speed at coordinate $r$ is $|r|/a$, so the endpoints have the null motion of a free string endpoint. The induced worldsheet metric degenerates there; the conserved densities below have finite limits, so the endpoint degeneracy causes no divergent charge.

In these coordinates ${\Pi }^{\tau \mu }=T{\dot{X}}^{\mu }$. The target-space Noether charges of a string give the energy, momentum and angular momentum:Since this is the centre-of-momentum frame, the rest mass is $M=E$. Eliminating $a$ proves the free-ended straight-string Regge relation:A spacetime Lorentz transformation of the solution changes its frame, not this rest-frame relation.

There is also a folded closed string version: use the same formula over $0\le \sigma \le 2\pi$. The spatial segment is then covered twice, with folds at $\sigma =0,\pi$, and the closed-string mode expansion is periodic. Each charge is twice its open string value, soThe folds also have degenerate induced metric and are understood through the finite canonical densities or a limiting smooth motion. The two classical Regge trajectories have different slopes because the closed string has two coincident branches. No quantum string intercept enters this classical calculation.

Use target signature $(-,+,\dots ,+)$ and worldsheet signature $(-,+)$. The Polyakov action has local worldsheet diffeomorphism and Weyl invariance, together with global target-space Poincaré symmetry. Under a worldsheet coordinate change $\sigma \mapsto {\sigma }^{\prime }$, the embedding coordinates are scalars and the worldsheet metric is a tensor:A Weyl transformation acts as $g_{ab}\mapsto e^{2\rho (\sigma )}{g}_{ab}$ and leaves $X$ unchanged. In two dimensions $\sqrt{-g}$ scales by $e^{2\rho }$ and $g^{ab}$ by $e^{-2\rho }$, so the action is invariant. Target translations act by $X^{\mu }\mapsto X^{\mu }+a^{\mu }$, and Lorentz transformations by $X^{\mu }\mapsto {\Lambda }^{\mu }{}_{\nu }{X}^{\nu }$ with ${\Lambda }^{\mathsf{T}}\eta \Lambda =\eta$; they leave $g$ unchanged. Infinitesimally the latter have $\delta X^{\mu }={\omega }^{\mu }{}_{\nu }{X}^{\nu }$, where ${\omega }_{\mu \nu }=-{\omega }_{\nu \mu }$.

Locally, the two diffeomorphism functions put the metric into the form $g_{ab}=e^{2\rho }{\eta }_{ab}$, and Weyl invariance removes the conformal factor. In this conformal gauge, the embedding coordinates are free scalar fields and satisfyThe metric equation must still be imposed after gauge fixing. Varying the ungauged metric gives the worldsheet stress-energy tensor constraintWith a dot and prime denoting the two coordinate derivatives, the independent Virasoro constraints areequivalently $(\dot{X}\pm X^{\prime }{)}^2=0$. Local conformal gauge leaves residual conformal transformations; on general worldsheet topologies it also leaves global moduli, rather than removing all metric information.

For target translations, differentiation of the Lagrangian with respect to ${\partial }_a{X}_{\mu }$ gives the Noether currentThe embedding equation is ${\partial }_a{j}^{a\mu }=0$. In conformal gauge, $j^{\tau \mu }={\dot{X}}^{\mu }/(2\pi {\alpha }^{\prime })$, so integration over the closed string givesPeriodic boundary conditions make its time derivative vanish. For an infinitesimal Lorentz transformation, the antisymmetric coefficient of ${\omega }_{\mu \nu }$ gives the current $j^{a,\mu \nu }=X^{\mu }{j}^{a\nu }-X^{\nu }{j}^{a\mu }$. Its divergence vanishes because the remaining product of embedding derivatives is symmetric in $\mu ,\nu$. Thus the Noether charge isThese are the target-space Noether charges of a string.

For the circular solution take $R>0$, and let ${\mathbf{e}}_r=(\cos \sigma ,\sin \sigma )$ and ${\mathbf{e}}_{\sigma }=(-\sin \sigma ,\cos \sigma )$. Its derivatives areEach spatial coordinate has equal second derivatives in $\tau$ and $\sigma$, while the time coordinate has both second derivatives zero. Furthermoreso both the field equations and the Virasoro constraints hold, including the collapse instants.

At target time $t=X^0=R\tau$, the string is a circle in the $X^1{X}^2$ plane of radius $R|\cos (t/R)|$. It contracts from radius $R$ to zero and re-expands; the negative cosine reverses material labels by a half-turn rather than creating a negative radius. The spatial configuration repeats after target time $\pi R$. This is a pulsating circular string. Its induced metric degenerates at collapse, but the conformal-gauge embedding and constraints give its continuation.

The integrals of $\sin \sigma$ and $\cos \sigma$ vanish, so the momentum isThe spatial velocity is radial, giving zero angular momentum density in the plane; all boost-charge terms integrate to zero as well. Hence $J^{\mu \nu }=0$ for every pair of indices.

At maximum expansion, all string elements are instantaneously at rest, and the length is $L_0=2\pi R$. Thus the rest mass per unit length, equal to the string tension, isAt other times the instantaneous circumference is $L=2\pi R|\cos \tau |$, and the elements have speed $|\sin \tau |$. Their Lorentz factor is $1/|\cos \tau |$, so the energy remains $TL/\sqrt{1-v^2}=2\pi TR$. The rest-energy density is $T$; the total rest-frame energy divided by the changing circumference is not constant. This distinguishes the requested intrinsic mass density from the larger energy density of moving string elements.

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.