Well, hey. I don't see my site show up here that often. If anyone's got comments, questions, or suggestions, I'll try to stop by and respond at some point.
I remember needing to relearn Lagrange multipliers several times throughout undergrad and grad school. Each time I needed a refresher, I ended up back at this "cow and the river" example. For some reason it always seems to map nicely back to whatever the current problem I am focused on is.
One application of Lagrange Multipliers includes deriving the micro, standard, and grand canonical ensembles from statistical physics [0].<p>These can be used to simulate physical systems like magnets and particle fields.<p>The Lagrange Constraints are used to constrain things like probability, energy, and particle numbers. What is constrained accounts for the difference between the ensembles listed above.<p>The derivation of the Support Vector Machine also involves Lagrange Constraints as well!<p><a href="https://en.m.wikipedia.org/wiki/Statistical_mechanics" rel="nofollow">https://en.m.wikipedia.org/wiki/Statistical_mechanics</a>
Lagrange’s own intuition into the multipliers was based on a purely formal construction of a differential equation that would incorporate the data about a point at equilibrium subject to constraints as well as about the constraints themselves. (I find the idea of “the virtual motion” in statics, where there is no motion, fascinating.) Here’s the story:<p><a href="https://abel.math.harvard.edu/~knill/teaching/summer2014/exhibits/lagrange/genesis_lagrangemultpliers.pdf" rel="nofollow">https://abel.math.harvard.edu/~knill/teaching/summer2014/exh...</a>