The article claims "In Lagrangian Mechanics you minimize the total energy of a system to find its motion". But if you actually look into it, you derive Lagrangian Mechanics by minimizing (or maximizing) the _action_. Action in this context is defined as the integral over the Lagrangian, or in other words the _difference_ between kinetic and potential energies (not their sums). The author knows that because later he states "All physical processes take the path that minimizes total action." and gives the corresponding mathematical expressions.

Also the statement about the Lagrangian being representation independent is misleading. The formalism stays the same and this is great for picking "suitable" coordinates that makes solving the system easier, but how the Lagrangian looks and how the equations of motion for the coordinates look can be quite different. This is actually what makes Lagrangian mechanics so powerful. You can transform coordinates (possibly several times) to get the Lagrangian into a simple shape and get equations of motions you can actually solve. You have transformed some of the difficulty of solving differential equations into the difficulty of finding natural coordinates, something humans tend to be better at.

The point that the multipliers \lambda_i that appear when modeling constraints should be names "Lagrange multipliers" has already been raised.