### Description of Course Goals and Curriculum

At the most basic level, PHY 205 is designed to introduce students to a new mathematical way of looking at physics. This course focuses on teaching the Lagrangian and Hamiltonian formalisms of physics, and these are introduced in the context of Newtonian mechanics, so students are already familiar with the underlying physics and can focus more on the new mathematical framework that is being introduced. These formalisms are more abstract than the Newtonian formulation, and so there is less frequently an associated physical “thing” with every mathematical concept. In sacrificing intuitiveness, though, students gain a greater flexibility in the kind of physical systems they can solve/understand and can tackle more complicated phenomena with greater ease.

Within the framework of Lagrangian and Hamiltonian mechanics, there are a few major areas of focus that this class pursues. One focus is on exploiting the relationship between symmetries and conservation laws, and deriving mathematical statements about conservation from both qualitative and quantitative observations about the symmetry of a system. Another focus is to give students greater comfort with harmonic oscillators, and especially teaching the usefulness of approximating a variety of different potentials as simple harmonic oscillators. In a related way, this course gives students a better understanding of Taylor series expansions as applied to physics and potentials, and gives students the tools to use Taylor series in simplifying lots of complicated problems. This course also teaches students how to expand physical laws into different inertial frames, including rotational and other non-inertial reference frames, thus giving students tools to do a wider variety of things and with a lot more mathematical ease than when using the Newtonian formulation of mechanics.

For the most part, this course introduces any necessary or useful mathematical concepts needed, especially those not covered in prerequisite physics classes. However, there are some times when the professor might assume more knowledge on a (often mathematical) topic than a large portion of the class actually has, and in these instances it is often good to stop and ask the professor to explain that topic in more detail, especially as it applies to the physics being presented in that lecture. This is especially true of concepts like Taylor series and Taylor expansion, with which many physics students are often only somewhat comfortable; differential equations, much of which has already been taught in other prerequisite physics classes but which still might be challenging if a student hasn’t had a differential equations class; Fourier series, which are covered in adequate detail but perhaps not on a level to make everyone fully comfortable with Fourier series; and special relativity, which isn’t introduced until the very last week, but which may be difficult to follow if the student doesn’t have some exposure to it previously. These are all topics that, while not necessary to fully understand going into the course, may present a bit more of a challenge if a student hasn’t had prior or concurrent exposure to these concepts in another class.

### Learning From Classroom Instruction

The nature of this material is that it is best understood by learning a concept and then immediately applying that to one or multiple examples. This is how the professor structures lectures, teaching a general concept and then doing specific examples on the board. There is a lot to go through, though, and so the pace is extremely fast; you cover about a chapter/topic per week. One helpful method of note taking in this environment is to try and get down all the information, spoken and written, in some form into notes, even if they don’t seem super coherent or you aren’t quite following the flow of a certain example. This is probably easier in pencil/paper form, unless you’re able to type up equations extremely quickly. That way, you can go back later and go over a section again if you didn’t completely follow it in lecture, or can look again at an example that is particularly relevant to a specific homework problem. In fact, going back over the notes after class is a really good way to see what parts you did and didn’t follow during the lecture, and to either then reread your notes at a slower pace and understand what you missed the first time around, or to figure out what things you need to ask about in the next lecture or in office hours. If you can’t get everything you want to down in your notes, at least writing down all the steps in any and every example will be extremely useful.

This course tends to have a wide variety of students in it in terms of their physics and math backgrounds. As such, the professor will often assume a certain useful physical principle or mathematical tool is familiar to everyone and will use it. Often he will check with the class to make sure everyone is already familiar with the newly introduced concept or symbol, but sometimes he won’t, and so it’s always good to speak up if something isn’t clear, especially if this is the first time it’s been introduced in this class. The professor is quite receptive to this kind of request for clarification and does a very good job of going over these new concepts if he is asked. Chances are if one person in the class hasn’t seen a concept, a large portion hasn’t, and the professor is very good at backing up to explain things if he realizes that there’s actually a need to. Often times students will be expected to be familiar with these concepts on homework and exams, so if you’re not, the best way to remedy that is to ask the professor to explain it in class. Examples of such concepts include Fourier series, Einstein summation notation, the Levi- Civita symbol, the Kronecker and Dirac delta symbols, Taylor expansions, etc.

The lectures complement the readings well, in that doing both will help your understanding a lot more than doing just one or the other. Lectures tend to go quite quickly, and while they are chock full of excellent information and examples, it’s often hard to truly get everything you need out of lecture. Ideally doing the readings before lecture would be preferable, so you’re already familiar with the concepts and can pay special attention to clarifications that weren’t in the book, or applications the professor goes over. However, reading after the lecture can also be useful because it can fill in the gaps in what you inevitably will have missed during the lecture, as well as possibly some details that simply didn’t get covered in lecture. In theory all the necessary material is in lecture, but for a truly solid understanding of the material, the book should be used to supplement the lecture. The book can also provide a source of questions to ask in the lecture, and since there is a lot of new material, being prepared with questions can be extremely helpful.

### Learning For and From Assignments

Assignments tend to do a rather good job of hitting on every important point from that week, and as such they can be quite challenging, especially if you’re not completely comfortable with all the concepts or are behind in your learning for that week. These are best done by starting early, ideally the day of the last lecture that covers relevant material, and spreading out the work. Working with other people for at least part of the time is an excellent idea, because often others will see a shortcut, simplification, or even way to start the problem that you hadn’t considered, and having exposure to these multiple strategies is really useful he next time you need to do a pset or when on an exam. Going to office hours, especially for the professor (but also for the TA’s) can also be super useful in everything from getting started on a problem (“How do I even begin to approach this?”) to understanding a concept (“I’m doing this but I’m not sure why”) to getting past a mathematical difficulty (“Is there a trick to this?”). Making a weekly habit of going to office hours wouldn’t be a bad idea. Also utilizing your notes, especially examples done in class that you wrote down, can be really useful for seeing a certain useful mathematical trick or way to approach a problem. The book can also be a good source of examples that might be able to push you in the right direction.

The exams tend to be structured as only 3 or so conceptual problems, but broken into multiple parts. They stress taking concepts that the course teaches and applying them in slightly different ways than you have before. As such, a good understanding of the underlying concepts, and not just a methodology for approaching problems, is really important for preparing for the tests. Going through practice problems and understanding WHY a certain approach is best, or why this equation is applicable in this context, is really useful preparation.

### External Resources

Occasionally simply searching the web for a different introduction to a topic can be useful. For example, for getting a hang of the equations of motion in a rotating reference frame, it was useful to get a fresh perspective on how to understand what to do and how to think about it by going to the internet (whatever you can find; often web pages from similar classes at other universities are most useful).

### What Students Should Know About This Course For Purposes Of Course Selection

The nature and purpose of this course is to give you a completely different mathematical framework with which to view mechanics and physics in general. As such, it has a bit of a steep learning curve at first, at the beginning can feel rather daunting. If you can push through this phase, or even better devote yourself particularly strongly during this phase, you’ll be able to really appreciate the underlying math and physics that is actually very cool and rewarding. It is a very demanding course that forces you to look at physics in a rather different way, but it also opens up the door to countless really awesome insights and realizations, as well as serving as the foundation of most upper-level physics.