Mechanical constraint

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

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.