PAUSE THE VIDEO IF YOU NEED MORE TIME!
This (long) video discusses:
1. Gauge Invariance=The invariance of the Equations of motion under certain transformations of the Lagrangian.
2. Coordinate Independence= The invariance of the Equations of motion under invertible differential coordinate transformation(known as diffeomorphisms).
3.The Hessian Condition= The condition in which we are able to determine the acceleration from knowledge of generalized coordinates and velocities.
4. We discuss the Hamiltonian Function and energy and their Relationship.
That's not gauge invariance, that's just normal group invariance.
miklysenko 7 months ago