I wondered if I had to upload this question on Physics StackExchange or on the Math one and I decided that it is still about mathematics so it's better to upload here. So please excuse me even if I am in the wrong forum.
I am a high school student. I was skimming through a mechanics book and found something called "Lagrangian mechanics". Here a quantity Lagrangian is defined. That's the only thing I know about it. I googled and found that it is mechanics using Lagrange's methods.
Also, I heard about the word Lagrangian multiplier but I don't know what exactly it is. I thought Lagrangian mechanics has something to do with this multiplier. I also heard from my economics class that Lagrangian multipliers are extensively used for the purpose of optimisation.
Q1. Do Lagrangian mechanics and Lagrangian multipliers just share the same name but are fundamentally different? Or not? Q2. Also I heard that "calculus of variation" is a method of optimisation. What's the relationship between it and Lagrangian multipliers?
Thanks in advance.
The method of Lagrangian multipliers Is useful when you want to find minimum or maximum of functions of manu variables.
The Lagrangian function Is a function very useful in physycs, wich describe the motion of body. It is definied as $$ \mathcal L = \mathcal T- \mathcal U \tag 1 $$ where $ \mathcal T$ is the cinetic energy and $\mathcal U$ is the potential energy.
They have nothing in common, except for the name Lagrange.
Edit
As pointed out by Lutz Lehmann if we want to minimize (or maximize) the function $f(x)$ subject to the constraints given by $g(x)=0$, then we define the Lagrangian as $$ \mathcal L = f(x) + \mu g(x) $$ (Note the analogy with the Physical definition in $(1)$)
The coefficients $\mu$s are called Lagrange multipliers