Lagrange multiplier dependent on time

Question

At the moment I am following a course on variational methods for mathematicians. Last week we derived the Euler-Lagrange equations for a functional under a constraint. In this derivation we found that the Lagrange multiplier may depend on time. However, I could not remember having ever seen an example of a Lagrange multiplier dependent on time during my physics courses. Does anyone have a nice (mechanical) example?

Answer 1

I don't have an example of one that depends on time time, but I have one for something similar. Suppose you have an elastic rod confined to some surface. The energy is the integral of the squared curvature

$$ H = \int \frac{K^2}{2} ~\mathrm ds $$

where the integral is over the arclength of the curve. We enforce the confinement to the surface by adding in something like

$$ H = \int \left[ \frac{K^2}{2} - \vec{\lambda}(s) \cdot (\vec{Y}(s) - \vec{X}(u^\alpha (s)) \right] ~\mathrm ds $$

where $\vec{Y}$ describes the curve and $\vec{X}$ describes the surface. The Lagrange-Multiplier $\vec{\lambda}$ is actually the force pushing the rod onto the surface, and is in general different at every point on the curve.