Originally it was likely created to study constrained mechanical systems where you want to use some "custom convenient" variables to parametrize things instead of x, y, z. Classical examples include:
However, people later noticed that the Lagrangian had some nice properties related to Lie group continuous symmetries.
Basically it seems that the easiest way to come up with new quantum field theory models is to first find the Lagrangian, and then derive the equations of motion from them.
Furthermore, given the symmetry, we can calculate the derived conservation law, and vice versa.
And partly due to the above observations, it was noticed that the easiest way to describe the fundamental laws of particle physics and make calculations with them is to first formulate their Lagrangian somehow: why do symmetries such as SU(3), SU(2) and U(1) matter in particle physics?.
- https://physics.stackexchange.com/questions/254266/advantages-of-lagrangian-mechanics-over-newtonian-mechanics fucking closed question...
- https://www.youtube.com/playlist?list=PLX2gX-ftPVXWK0GOFDi7FcmIMMhY_7fU9 "PHYSICS 68 ADVANCED MECHANICS: LAGRANGIAN MECHANICS" playlist by Michel van Biezen.High school classical mechanics material, no mention of the key continuous symmetry part. But does have a few classic pendulum/pulley/spring worked out examples that would be really wise to get under your belt first.
- http://www.physics.usu.edu/torre/6010_Fall_2010/Lectures.html Physics 6010 Classical Mechanics lecture notes by Charles Torre from Utah State University published on 2010,
- Classical physics only. The last lecture: http://www.physics.usu.edu/torre/6010_Fall_2010/Lectures/12.pdf mentions Lie algebra more or less briefly.