The principle of least action: a more elegant mechanics
wherenothinggathers.blogspot.comIf I remember nothing else from Physical Mechanics 1 -- and I probably won't -- I will always remember that the calculus of variations and the principle of least action are fucking dark magic.
It gets even better with Hamiltonian mechanics, canonical transformations, and Poisson brackets, just wait! Especially neat is how easy it is to connect such abstract formulations of mechanics directly to quantum mechanics. I love seeing reflections of the same underlying principle in different subfields - it's elegant and inspiring!
My recollection is of a deep irritation, like, "You people have me dorking around with all these hopeless special cases, when all along we could have been doing this? For fuck's sake, why?"
The action integral isn't solvable. You need to go to the special cases to get anything done. It is like saying energy is always constant. Sounds profound, but actually computationally useless on its own, since the invariant is broken frequently, until we go find something else to also call energy (gravitational potential, kinetic, electric potential, heat, mass! , etc)