What are the immediate advantages of the Lagrangian formulation compared to the Newtonian formulation to give high school students a little connection to what they'll find in universities? I can't use derivatives, integrals, ODE, ecc... but very few elementary concepts.
Are there advantages to solving dynamic or kinematic exercises? What could they be? Often in classical electrodynamics special lagrangian are used to obtain an expected result.
Lagrangian Dynamics already includes constraints, that need to be added by hand in the Newtonian formulation. Furthermore, the Lagrangian formalism is suitable for fields, see the Euler Lagrange equations.
Classical mechanics was given birth with the publication of Philosophiæ Naturalis Principia Mathematica by Sir Isaac Newton in 1687. It finally laid to rest Aristotle’s view of motion and was a basic framework for the physics to come over the following century. The Principia contained Newton’s universal law of gravitation as well as Newton’s three laws of motion. Together, they connect the Earth with the Heavens in one construction.
The only disadvantage to Newton’s laws is they are written in terms of vector quantities, quantities that depend on direction. This makes the mathematics behind them a bit of a hassle at times and arguably less elegant. A couple of years after the publication of the Principia, Gottfried Wilhelm von Leibniz (the German mathematician that invented calculus independently from Newton) began to voice opinions of a scalar quantity he had noticed which he called vis viva. This scalar would eventually become known as kinetic energy $\displaystyle KE=\frac{1}2mv^2$. The idea of scalar quantities was opposed by Newton for quite some time because he felt it was inconsistent with his conservation of momentum.
In 1788, Joseph Louis Lagrange published “Analytical Mechanics” where he derived his equations. These equations were contrasted from Newton’s because they were formulated entirely in terms of scalar quantities.
Lagrangian mechanics is widely used to solve mechanical problems in physics and when Newton's formulation of classical mechanics is not convenient. Lagrangian mechanics applies to the dynamics of particles, while fields are described using a Lagrangian density. Lagrange's equations are also used in optimization problems of dynamic systems.
It's all about the way that you're solving the problem, using the Newtonian mechanics you do the projections of the vectors after using the fundamental theorem of dynamics $\vec{F}=m\vec{\gamma}$ and it's all basic and simple, but solving a problem in which there's a lot of constraints the situation begins to be complex so you better use your Lagrangian, and it may be suitable for electrodynamics as you said special relativity, GR, hydrodynamic, etc.