PH408 - Analytical Mechanics and Special Relativity - Spring 2021 - 3 credits

Contact Information


This will be an advanced course in the description of motion. We will begin by reviewing basic Newtonian mechanics. We will then take a detour into the theory of special relativity, which describes the consequences of the fact that the speed of light is the same in all inertial reference frames, forcing times and distances between events to depend on the speed of the observer. Relativity is a necessary consideration when analyzing motion at very high speeds (compared to the speed of light). We will also introduce a mathematical framework known as the calculus of variations, which allows a function to be optimized in a way that maximizes or minimizes its integral. We will use the calculus of variations to motivate Lagrangian mechanics, which comes from the principle that the equations of motion of a system must minimize the integrated difference between the kinetic and potential energy. Lagrangian mechanics dramatically simplifies the solutions of many problems. We will apply these sophisticated mathematical tools to analyze phenomena such as oscillations and orbits.

Tentative schedule

A rough schedule can be found here.


Students in PH465 should already have taken the PH304 sequence (introductory physics with calculus) and MT463 (Differential Equations).

This course is like the first semester of general physics, but much more mathematically oriented, so you should have mastered the material taught in PH 304A. We will often need to solve differential equations, so you are expected to have taken MT 463 or, at the least, to be taking it concurrently. Calculus will be used liberally in all aspects of the course, and it will not be significantly reviewed or re-taught. You should be comfortable using and interpreting derivatives and integrals before starting any upper division physics courses. In addition, it would be ideal, but not required, to have some experience with vector calculus (as taught in MT 360C) and linear algebra (MT 415). We will need to use some techniques that are covered there, but we will introduce those techniques in this course.

We will occasionally use computational tools to assist us. Students should be comfortable using at least one of the major operating systems (Windows, Mac OS, or Linux). Past exposure to a programming language (e.g., Python, etc) will be helpful but is not required.

All upper division physics courses are interrelated, so we will occasionally mention topics related to quantum mechanics and electromagnetism, but upper division courses on these topics are not required to take this course. All necessary information on these topics is provided in our textbook or in your introductory textbooks.

COVID-19 Policies

The format of this course will be set entirely at my discretion using the guidance from the University administration, according to the best current scientific understanding of the pandemic. The format may change as the pandemic evolves, and the safety precautions we observe when meeting may change as well. You are required to follow all safety policies set by me and/or the university at all times. There will be no tolerance for recklessness, and repeated violations will result in failure and removal from the course.

As of the beginning of the semester, properly fitting masks are required of everyone in the class at all times without exception. I strongly recommend well-fitted N95 masks; cloth masks are not sufficient to protect against the transmission of the Omicron variant. Vaccines and boosters are required of all students who have not received an exemption from the university.

My Classroom Rules – A Commitment to an Inclusive Classroom

More than anything else, this class is a community that values the diversity of experience that each of us brings to our shared journey in physics. My first priority is the creation of an inclusive classroom - one where everyone feels comfortable, intellectually challenged, and able to speak up about your ideas, struggles, and experiences. Our classroom, as well as its virtual extensions, must foster respect, civility, and an ability to listen. I’ll do my best to reach these goals myself, but I’ll liekly err at times. Please let me know if there is anything I can do to make our classroom a more inclusive and productive learning environment. If anyone in our community is violating our commitment to inclusivity, please let me know. If I am violating this commitment, I welcome criticism so that I may learn to do better, and I encourage you to let me know. If you don’t feel comfortable talking to me directly about it, you are also welcome to raise the issue with our department chair, Prof. Stacy Chamberlin (

You should expect respect from me and your classmates, and you should also endeavor to be kind and respectful to everyone else. That means a lot of things, including:

We all make mistakes with others; the important thing is that we learn from them and use it to help others in the future.

Confidentiality and Mandatory Reporting

College students often experience issues that may interfere with academic success such as academic stress, sleep problems, juggling responsibilities, life events, relationship concerns, or feelings of anxiety, hopelessness, or depression. As your instructor, one of my responsibilities is to help create a safe learning environment and to support you through these situations and experiences.

I also have a mandatory reporting responsibility related to my role as a university employee. It is my goal that you feel able to share information related to your life experiences in classroom discussions, in written work, and in one-on-one meetings, so I will seek to keep information you share private to the greatest extent possible. However, under Title IX and the Clery Act, I am required to share information regarding sexual misconduct, relationship violence, or information about criminal activity on or near Regis’s campus with the university.

If you wish to speak about these issues confidentially, the Office of Counseling and Personal Development (OCPD) can help. OCPD is located in the Coors Life Direction Center, Room 114 and can be contacted by phone 24/7 at 303-458-3507. For more information, see

Important Notes on Expectations and Success

What You Should Expect

PH 408 covers material at a high level of conceptual and mathematical sophistication.

PH 408 is a challenging, upper-division physics course. Unlike more introductory courses, you are fully responsible for your own learning. In particular, to some degree, you control the pace of the course by asking questions in class. This means that if you don’t understand something, it is your responsibility to ask questions. Attending class gives you an opportunity to ask questions.

How to Succeed in this Course

The topics that we will cover in PH~465 are among the greatest intellectual achievements of humans. Don’t be surprised if you have to think hard and work hard to understand the material. You can perform very well in this class if you do the things necessary to learn physics:

Academic Integrity

Unfortunately, there has been a large spike in dishonest academic actions during the pandemic, so I need to spend some time being blunt about it.

Be aware: Any indication of academic integrity violations is grounds for immediate failure of the course. An indicent report will be filed with the dean’s office, and you will immediately fail, along with whatever other sanctions your college or program imposes.

So what does academic dishonesty look like? In the modern internet-connected world, it is not possible to enumerate all types. In summary, though, you are expected to be submitting your own work, and if you aren’t, it is academic dishonesty. If you are looking at a solution by a different person to a problem that I have assigned, then you are committing academic dishonesty. If you are copying any solution and misrepresenting it as your own, whether you got it from the internet or another student, it is academic dishonesty.

Like I said, there are too many forms of this sort of thing to list, but I’ll mention a few key ones.

This list is not all inclusive. Remember: Your work is expected to be your own. Though this can sometimes be a difficult subject, it is within your grasp and I am here to help. Even though the class is hard, I also design the grading so that it is hard to fail if you try. I want to help you succeed, but you must be honest with me if I am to do that. Do not gamble your whole career on one test or assignment.


One of the most important aspects of success in fields of high levels of intellectual difficulty is self-care. It is also one of the most neglected.

Course communication

We’ll be using WorldClass for grade distribution and other digital communications. It will host homework and solutions. We’ll also use the discussions section for homework help, and you should feel free to chat on it, vent, organize study sessions, etc. I intend to cultivate a rather informal learning community, so you should feel free to make it your space, but please also remember that it is an extension of our classroom at Regis, so you should behave in compliance with the student handbook.

You are unlikely to succeed in this class (or any other physics endeavor) on your own; you should make it a priority to support your peers and to accept their help. Collaboration is not only encouraged, it is implicitly required. I strongly suggest that you all frequently get together in person at times in addition to our scheduled meetings to discuss the material of this course.

In all cases, collaboration should be done in a productive, positive, humble, and honest way. You must do so in a way that is both productive for your learning and within the guidelines of the Academic Honor Code. This means that you should never be simply copying or sharing solutions - to do so is both useless for learning and an academic integrity violation. Seriously, don’t cheat, but seriously, help each other. This also means that you should be kind in offering assistance. Be generous, but do not grandstand or belittle. Listen as much as you talk, and always be asking yourself if your conversational approach is truly in service of truth-seeking and care for others.

Course Materials

Required book:

Course Activities


Reading is an essential part of PH 408! Reading the text before class is very important. Our class meetings are to clarify your understanding and to help you make sense of the material. I will assume you have done the required readings in advance and I will not simply repeat everything you should have read! Our book is fairly well written and readable - it will make a huge difference if you spend the time and effort to carefully read and follow the text. The calendar has the details on reading assignments.


You are expected to attend and fully participate in every class, both for your own learning and out of respect for your classmates. This course will be conducted in a “partially-flipped” model where you are expected to prepare for most class days by carefully reading the textbook. That preparation will enable us to use the class meetings for activities intended to deepen your understanding of the material that you have read about. Class can become a time to discover the questions that you have about the concepts and techniques that we are discussing, and to get those questions answered. In general, I expect that you will attend and participate fully in every class. A small part of your grade will be based on completion of in-class activities. They will be graded only for completeness and apparent effort, not for correctness.


There will be two in-class midterm exams and a final exam. All exams are cumulative, although midterms will primarily focus on the material introduced since the last midterm. Details about exam format will be given in class.

All exams are cumulative of everything covered so far in the course, as well as mathematics through calculus and the introductory physics sequence. Physics is inherently cumulative, always.


There will be a homework due approximately each week. Homework is arguably the most important part of a physics class – it is where you learn to use the concepts. You should start homework early and work on it consistently. You should treat the assigned problems as the bare minimum number of problems that one could plausibly need to solve to learn material; you should almost certainly be solving more problems than what is assigned for each topic.

You are encouraged to collaborate on difficult homework problems, but you must do so in a way that is both productive for your learning and within the guidelines of the Academic Honor Code. This means that you should never be simply copying or sharing solutions - to do so is both useless for learning and an academic integrity violation. Seriously, don’t cheat. You have no idea how easy it is to spot for your physics professors - the only question is when they decide to enforce it. Don’t gamble your whole academic career on saving some time on a homework problem. Instead, first take some time alone to attempt each problem and create your own ideas as to how they might be solved. Once you’re stuck, get together with classmates to brainstorm, talk through the problems, and work through a viable solution strategy. Finally, separate from the group, and write up a solution on your own, filling in the details. Remember, if you can’t solve similar problems on your own without help, then you haven’t learned the material and will not succeed on the exams or in real-life applications. Test yourself and, most importantly, be honest with yourself! For all assignments, the work you turn in must in the end be your own: in your own words, reflecting your own understanding. If, at any time, for any reason, you feel disadvantaged or isolated, contact me and I can discretely try to help arrange study groups.

You should cite any and all external assistance that you receive on any work - in fact, you should do this everywhere in life! If Angeline Kumar and Gunnarr Skuld helped you with part of a problem, then add a note to that effect (e.g., “Gunnar and I came up with the main idea of the solution together, and Angeline pointed out to me that we were counting the degrees of freedom incorrectly.”). If everyone, when hopelessly stuck, looked at the Wikipedia page for the “Ideal Gas Law” and happened to see a major piece of the solution, cite it explicitly. We all need help with physics sometimes, but we also need to both acknowledge that help and build on it with original thinking of our own.

In most cases, late homeworks will not be accepted. If you have an illness, family emergency, or similar excused reason for not completing an assignment, then you will be excused from that assignment. It is your resonsibility to go back and work the problems as soon as you can.

Computational Homework Problems

There will be some use of computation in this course on homework problems. You will need to use Python (through Jupyter notebooks). You do not need any computational experience for this course as you will learn some fundamentals early on and keep using them throughout the course. Python is used across the sciences as well as private industry, so learning it will serve you well in your future work, no matter what it is. Artists use Python, teachers use Python, data analysts use Python, sports teams use Python, and physicists use Python. I suggest downloading the Anaconda distribution of Python as it comes with all the packages you will need to get up and running with Jupyter notebooks, but you are welcome to use any other method of running Python code as well. In particular, you might find something like Google Colab useful, which can interface with Github, Google Drive, or a local storage drive.

Everyone will submit code via GitHub As you become involved with computational or analytic activities in the professional world, you’ll find that people expect you to manage your code and documents professionally, too. The standard way of doing so is with git repositories (“repos”), often stored online on Github for easy sharing, distribution, and collaboration. You should make a Github account if you do not already have one; you are welcome to use a throwaway email address if you do not want to share your primary one. (However, Github is a very standard tool – if you think you might have involvement with any technical field or hobby in the future, I recommend making an account that you’ll be able to easily use for many years in many contexts, so use an email to which you’ll retain access and choose a username that you wouldn’t mind showing to a potential employer.)

If you get stuck with the technological aspects of the class, don’t be afraid to check the guides that I’ve posted here, ask for help on WorldClass, come see me, or Google it (as should usually be your first recourse).

Grading and Exams

Your course grade is determined as shown:

ActivityDatePercent of Grade
Exam 1Wednesday, ?? /?? (tentative)15%
Exam 2Wednesday, ?? /?? (tentative)15%
Final ExamApril 29 @ 1:15p20%
Homeworkusually Fridays40%

Owing to the small class size and the inaccuracy of predicting grade distributions on upper division physics exam problems, the final mapping of letter grades to numerical scores will likely be revised at the end of the semester. However, as a rough guide, you should expect score distributions approximately as follows:

Letter GradeDescriptionNumerical Grade
AOutstanding scholarship.>88
A- 85 – 88
B+ 82 – 84.99
BStrong command of material.78 – 81.99
B- 75 – 77.99
C+ 72 – 74.99
CSatisfactory command of material.68 – 71.99
C- 65 – 67.99
D+ 62 – 64.99
DUnsatisfactory command of material.58 – 61.99
D- 55 – 57.99
FNo credit.<55

I reserve the right to assign higher grades than would result from this table according to my judgment of the difficulty of the assignments and exams. No final grades will be adjusted or curved downward relative to the above table under any circumstance. For example, if everyone earned above an 88%, then everyone would get an A, so it is in your interest to work together and help each other learn physics.

Further Policies

Additional syllabus policies, including all university-wide policies, are enumerated here.