Constraint (mechanics)

A mechanical constraint restricts the allowed configurations or velocities of a system and reduces its degrees of freedom.

A collection of Hamiltonian constraints is second class when its Poisson bracket matrix $C_{ab}=\{{\chi }_a,{\chi }_b\}$ is invertible on the constraint surface. These conditions eliminate canonical variables rather than generate a gauge symmetry. Their consistent reduced bracket is the Dirac bracket, $\{F,G{\}}_D=\{F,G\}-\{F,{\chi }_a\}(C^{-1}{)}^{ab}\{{\chi }_b,G\}$. The zero temporal canonical momentum of a massive Proca field and its massive Gauss law form such a pair; imposing an independent temporal canonical commutation relation would contradict the constraint.

A particle of mass $m$ constrained to the cone $z=-r$ is joined through the apex by a light inextensible ideal string to a hanging particle of mass $\sqrt{2}m$. The cylindrical radius is $r>0$ and gravitational acceleration is $g$. If the total ideal string length is $s_0$, the second particle has vertical coordinate $z_2=-s_0+\sqrt{2}r$. The ideal string and normal reaction exert no azimuthal force, so conservation of angular momentum gives $r^2\dot{\theta }=h$.

The first particle has kinetic energy $m{\dot{r}}^2+m{h}^2/(2r^2)$, while the second has kinetic energy $\sqrt{2}m{\dot{r}}^2$. Their combined gravitational potential energy is $mgr$ plus a constant. Hence conservation of energy gives the displayed equation, with effective potential $U(r)=gr+h^2/(2r^2)$. For $h\ne 0$, $U$ is strictly convex and has its unique minimum at $r=(h^2/g{)}^{1/3}$. The energy level has two radial turning points unless the motion is a circular orbit at that minimum.

The model maintains a mechanical constraint on the surface. For unilateral contact with one side of the cone, a normal reaction computed with a fixed outward sign must also have the physically allowed sign. A finite length of ideal string additionally restricts $r<s_0/\sqrt{2}$ before the hanging particle reaches the hole.

The holonomic constraint obtained by summing the straight and curved segment lengths of a taut massless inextensible string and setting the result equal to its fixed length $l$. Differentiation gives ${\sum }_i(\partial L/\partial q_i){\dot{q}}_i=0$, a relation between the permitted velocities. A slack string does not impose this equality on endpoint separation.

An ideal contact mechanical constraint supplying a normal reaction and no tangential friction. For a stationary constraining surface, the permitted velocity is a tangent vector, so the normal force has zero power and does no work. For a moving surface, normality alone does not imply zero power in the laboratory frame.

A point mass constrained to the fixed graph $y=y(x)$ has velocity $(\dot{x},y^{\prime }\dot{x})$. Under uniform downward gravity, its Lagrangian is $L=m(1+y^{\prime 2}){\dot{x}}^2/2-mgy$, so its canonical momentum is $p_x=m(1+y^{\prime 2})\dot{x}$, not the horizontal mechanical momentum. The Euler-Lagrange equation gives the equation in the title, and the conserved energy is $m(1+y^{\prime 2}){\dot{x}}^2/2+mgy$. A straight segment with constant slope $s$ has constant horizontal acceleration $-gs/(1+s^2)$; zero curvature does not imply zero acceleration. A cart held to a track satisfies this mechanical constraint only while the assumed contact or retaining mechanism remains effective.

A nonholonomic constraint is a nonintegrable restriction on allowed velocities: it cannot locally be replaced by relations among generalized coordinates and time. In contrast, a velocity equation $r\dot{\varphi }-R\dot{\theta }=0$ with constant radii integrates to a holonomic constraint $r\varphi -R\theta =C$. Rolling without slipping is therefore not automatically nonholonomic: planar rolling on a prescribed curve can be holonomic, whereas rolling with additional directional freedoms can impose nonintegrable velocity constraints. Unwrapped angles may be needed to record the winding of a rolling trajectory.

For a smooth stationary wire $z=x^2/(4a)$ and release from rest at height $h$, the normal reaction does no work. The energy equation is $[1+x^2/(4a^2)]{\dot{x}}^2/2+g{x}^2/(4a)=gh$. Integrating the travel time between turning points gives the displayed period. Its small-amplitude expansion is $T=2\pi \sqrt{2a/g}[1+h/(4a)+O((h/a{)}^2)]$. The mass cancels because both gravitational and inertial forces are proportional to it.

The common zero locus of constraint functions $C_a$ in phase space is the constraint surface. A physical trajectory lies on this surface, but off-surface identities are often needed when analyzing a constraint algebra.

A holonomic constraint is a relation $f(q,t)=0$ involving generalized coordinates and possibly time, rather than a genuinely nonintegrable velocity relation. Independent smooth constraints of full row rank define a local configuration submanifold. Ideal constraint forces are normal to its tangent virtual displacements and can be represented by Lagrange multipliers. A time-dependent holonomic constraint need not conserve mechanical energy.

A first-class constraint has Poisson brackets with all constraints that vanish on the constraint surface. For a regular constrained Hamiltonian system, first-class constraints generate gauge transformations; each independent constraint and its gauge direction remove one canonical pair.

For $n$ independent regular first-class constraints in a $2N$-dimensional phase space, imposing the equations removes $n$ dimensions and quotienting independent gauge orbits removes another $n$. Reducibility, singular strata or vanishing gauge directions can invalidate that elementary count. Closure of the constraint algebra is necessary but does not by itself establish regularity.

A first-class constraint generates an infinitesimal canonical transformation $\delta F=\{F,{ϵ}^i{\phi }_i\}$. For the phase-space action $I=\int (p\dot{q}-{\lambda }^i{\phi }_i)dt$, the multiplier transformation $\delta {\lambda }^k={\dot{ϵ}}^k-{\lambda }^i{ϵ}^j{C}_{ij}{}^k$ cancels the bulk variation when $\{{\phi }_i,{\phi }_j\}=C_{ij}{}^k{\phi }_k$. The remaining temporal boundary term vanishes for parameters that vanish at the endpoints.

The Poisson brackets of first-class constraints close on the constraints. When the constraints are independent and the coefficients are constant, the Jacobi identity for the Poisson bracket makes the coefficients structure constants of a Lie algebra.

The coefficients $C_{ij}{}^k(q,p)$ in $\{{\phi }_i,{\phi }_j\}=C_{ij}{}^k{\phi }_k$ are structure functions. A finite real span of first-class constraints is a Lie algebra when it closes with constant coefficients. General structure functions instead describe closure with phase-space-dependent coefficients, while still permitting canonical gauge transformations.