We look at three applications of Noether's Theorem: a pendulum, a double pendulum and a harmonic oscillator.
All three examples show how much easier it is to use Lagrange's method to find the equations of motion, rather than Newton's, simply because Newton's form involves second-order differential equations to solve.
This becomes increasingly true if the number of coordinates in a system is particularly large, or aren't the usual spatial coordinates - for example in General Relativity.
In both forms, the positions and velocities are given, along with a law of nature. For Newton, a law of nature means a force, but for Lagrange, a law of nature means a potential.
Knowing the potential means that we can immediately write down the Lagrangian (since the kinetic energy term in the Lagrangian is given by the positions and velocities) and the Euler-Lagrange equations.
The harmonic oscillator allows us to introduce Hamilton's form, which is described with respect to the coordinates and their conjugate momenta, rather than their simple time-derivatives.