Lagrange multipliers for constraints

Remarks

The vector $\sum_a \lambda_a \nabla_q f_a$ is the generalized constraint force; it is normal to the constraint set, so ideal constraints do no virtual work. The same equations are the Euler–Lagrange equations of the action of $L + \sum_a \lambda_a f_a$, varied in both $q$ and $\lambda$. This is the dynamical analogue of the Lagrange multiplier theorem for constrained extrema. Eliminating $\lambda$ with coordinates adapted to the constraints gives the Lagrange equations of the second kind.

Example: a planar pendulum of mass $m$ and length $\ell$ in Cartesian coordinates, with $L = \frac{1}{2} m (\dot x^2 + \dot y^2) - m g y$ and $f = x^2 + y^2 - \ell^2$. The equations are $m \ddot x = 2 \lambda x$, $m \ddot y + m g = 2 \lambda y$ and $x^2 + y^2 = \ell^2$. The force $2 \lambda (x, y)$ points along the rod, and its magnitude $2 \abs{\lambda} \ell$ is the tension.

Non-example: the constraints $f_1 = x^2 + y^2 - \ell^2$ and $f_2 = 2 f_1$ describe the same set, but their gradients are dependent. Then only $\lambda_1 + 2 \lambda_2$ is determined, so the multipliers are not unique.