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Replace the derivative of an isometric coordinate by an independent one-form $A$ and impose $dA=0$ with a Lagrange multiplier. In signature $(-,+)$ with ${ϵ}^{\tau \sigma }=1$, its equation is $V{A}^{\nu }={ϵ}^{\mu \nu }{\partial }_{\mu }\tilde{z}$. Eliminating $A$ from the full action, including the multiplier term, gives $\frac{1}{2}{V}^{-1}{\partial }_{\mu }\tilde{z}{\partial }^{\mu }\tilde{z}$. Omitting the multiplier term during substitution would give the wrong sign. This derives the block-diagonal metric inversion in the Buscher rules.

For a differentiable objective on an admissible real Hilbert space $H$, with $\Vert {\mathbf{u}}_0{\Vert }^2/2=E_0>0$, stationarity on the sphere in a normed vector space of fixed kinetic energy means that the projected gradient is normal to the sphere in a normed vector space. Thus $P_H\nabla J=\lambda {\mathbf{u}}_0$ for a real Lagrange multiplier. It follows by testing all tangent variations orthogonal to ${\mathbf{u}}_0$. The multiplier need not be positive; second-order conditions are still needed for a local minimum.

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.

An inextensible filament keeps the length of every material segment fixed. With material arc length $s$, its centerline obeys $|{\mathbf{r}}_s|=1$ and has unit tangent vector $\mathbf{t}={\mathbf{r}}_s$. A filament tension acts as a Lagrange multiplier for this constraint.

For independent unbiased estimators $X_i$ of a common mean with known positive variances $v_i$, the linear unbiased estimator with smallest variance has weights $w_i=v_i^{-1}/{\sum }_j{v}_j^{-1}$ and variance $({\sum }_j{v}_j^{-1}{)}^{-1}$. Unbiasedness requires ${\sum }_i{w}_i=1$. Minimizing ${\sum }_i{v}_i{w}_i^2$ by a Lagrange multiplier gives the weights, and the strictly positive diagonal Hessian matrix proves a unique global minimum. This conclusion requires no normality assumption.

An isoperimetric constraint fixes the value of an integral while optimizing another integral over functions. A constant Lagrange multiplier incorporates the constraint into the variational integrand, and the resulting Euler-Lagrange equation must be supplemented by the appropriate natural boundary conditions for a free endpoint. The name includes integral constraints more general than prescribed geometric length.

A Lagrangian relaxation moves selected constraints into the objective with appropriately signed Lagrange multipliers. Maximizing the relaxed objective over a larger easy set gives an upper bound for a constrained maximization problem. Optimizing that bound need not solve the original integer problem.

The two constraints are $P^2+T^2{X}^{\prime 2}=0$ and $P\cdot X^{\prime }=0$. They are first-class constraints associated with worldsheet diffeomorphisms. Their Lagrange multipliers are arbitrary gauge functions rather than propagating fields.

For a minimization problem with $g_i(u)\le 0$ and $h_j(u)=0$, the optimization Lagrangian is $L=f+{\sum }_i{\lambda }_i{g}_i+{\sum }_j{\mu }_j{h}_j$ with ${\lambda }_i\ge 0$ and unrestricted equality Lagrange multipliers. Its infimum over the original variable domain gives a dual lower bound. For a maximization problem, use $L=f-{\sum }_i{\lambda }_i{g}_i-{\sum }_j{\mu }_j{h}_j$ to obtain upper bounds. The Lagrangian sufficiency theorem combines a global extremum of this function with feasibility and complementary slackness. This is distinct from a Lagrangian density in variational physics.