# Intermediate Mechanics

### PHYSICS361

Newtonian mechanics at the intermediate level, Lagrangian mechanics, linear oscillations, chaos, dynamics of continuous media, motion in noninertial reference frames. Prerequisite: MATH 216 or equivalent (may be taken concurrently). One course.

### Syllabus

Newtonian dynamics, conservation laws, systems of many particles, ocsillatory motion, damped and driven oscillators, resonance, impulse response, coupled oscillators, Hamilton principle, Lagrangian mechanics, Hamilton's and Lagrange's equations of motion, Noether's theorem and conservation laws, central force problems, Kepler's planetary motion, scattering by central forces, non-inertial reference frames, dynamics of rigid bodies, inertia tensor, Euler angles and equations, motion of a symmetric top, dynamics of continuous media, identical coupled oscillators and continuum limit, wave equation and boundary conditions, wave packets, phase velocity, group velocity, dispersion, non-linear systems.

### Textbook

Marion and Thornton, "Classical Dynamics of Particles and Systems," 5th Edition.

### Informal Explanation of the Course

In early attempts to understand the fundamental pieces of nature and the rules that govern how these pieces interact, scientists explored two paths in parallel, that the basic objects in the universe were localized particles (or rigid groups of such particles) or that the basic objects were spatially extended entities similar to waves. PHYSICS 361 discusses the first of these points of view, how to describe nature in terms of the motion of particles or of rigid bodies. In contrast, PHYSICS 182, the core course on electrodynamics, discusses how to describe nature in terms of the dynamics of spatially extended fields, such as electric and magnetic fields. Experiments later showed that the fundamental particles in nature like a photon or electron act neither like particles nor like waves but like something else, which is discussed first in PHYSICS 264 and in more depth in the introductory quantum physics course, PHYSICS 464.

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**=m**a**). 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?)