The undergraduate mechanics course introduces new more sophisticated ways associated with the names of Hamilton and Lagrange to describe the motion of particles and rigid objects. These new ways have multiple advantages over Newton's three laws of motion. One is that vector forces disappear from the laws of motion (no F=ma). This turns out to greatly simplify many mechanics problems, which become easier to solve. The Lagrange formulation especially makes clear something that is difficult to deduce from Newton's laws, which is a deep connection between symmetry and the evolution equations for a system of particles. This connection later played a central role in the development of theories of subatomic particles. It turns out that quantum mechanics has remarkable mathematical similarities to classical mechanics (which is an important reason to take 361 before 464) and can also be expressed in a Lagrangian form. It is then possible to guess what are the laws of interaction of subatomic particles (such as the strong interaction) by guessing what symmetries might govern their behavior.
It is in Physics 361, via the Lagrange and Hamilton formulations of mechanics, that you get to understand particularly clearly why there are additive conservation laws for energy, momentum, and angular momentum but generally not for other quantities. These conservation laws turn out to reflect fundamental symmetries of space and time. For example, energy conservation is associated with the observation that when an experiment is carried out does not affect the results of the experiment, this is the symmetry of time translation invariance. Momentum conservation is associated with the observation that where an experiment is carried does not affect the results and is associated with the symmetry of space translation invariance, and similarly for angular momentum. (A puzzle for you to think about: are all conservation laws related to some symmetry? If so, what is the symmetry that corresponds to conservation of charge?)