How does the Lagrange multiplier method work for constrained systems?

[Curl]

Is anyone good with Lagrangian mechanics applied to constrained systems?

I had a question about the Lagrange multiplier method, maybe I should have posted it in this section.

https://www.physicsforums.com/showthread.php?t=550139

 

[atyy]

Are you asking why Lagrange multipilers are a way to do constrained minimization?

 

[Curl]

I'm asking how is the method of Lagrange Multipliers (which is used to find the extrema of level surfaces subject to constraints) used to find the extrema of the Action functional.

I thought I'd be more of a math question but nobody there is throwing me an answer.