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

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: https://abel.math.harvard.edu/~knill/teaching/summer2014/exhibits/lagrange/genesis_lagrangemultpliers.pdf https://abel.math.harvard.edu/~knill/teaching/summer2014/exh...