This fall I'm going to be teaching honors introductory mechanics to incoming undergraduates - basically the class that would-be physics majors take. Typically when we first teach students mechanics, we start from the point of view of forces and Newton's laws, which certainly parallels the historical development of the subject and allows students to build some physical intuition. Then, in a later class, we point out that the force-based approach to deriving the equations of motion is not really the modern way physicists think about things. In the more advanced course, students are introduced to Lagrangians and Hamiltonians - basically the Action Principle, in which equations of motion are found via the methods of variational calculus. The Hamiltonian mechanics approach (with action-angle variables) was the path pursued when developing quantum mechanics; and the Lagrangian approach generalizes very elegantly to field theories. Indeed, one can make the very pretty argument that the Action Principle method does such a good job giving the classical equations of motion because it's what results when you start from the path integral formulation of quantum mechanics and take the classical limit.
A major insight presented in the upper division course is Noether's Theorem. In a nutshell, the idea is that symmetries of the action (which is a time integral of the Lagrangian) imply conservation laws. The most famous examples are: (1) Time-translation invariance (the idea that the laws of physics governing the Lagrangian do not change if we shift all of our time parameters by some amount) implies energy conservation. (2) Spatial translation invariance (the laws of physics do not change if we shift our apparatus two feet to the left) implies conservation of momentum. (3) Rotational invariance (the laws of physics are isotropic in direction) implies conservation of angular momentum. These classical physics results are deep and profound, and they have elegant connections to operators in quantum mechanics.
So, here's a question for you physics education gurus out there. Does anyone know a way of showing (2) or (3) above from a Newton's law direction, as opposed to Noether's theorem and Lagrangians? I plan to point out the connection between symmetry and conservation laws in passing regardless, but I was wondering if anyone out there had come up with a clever argument about this. I could comb back issues of AJP, but asking my readers may be easier.