The Techniques of Theoretical Mechanics course introduces physics majors to ways of making sense of theoretical physics.

An example course schedule can be found here with activities, resources, and homework problems.

#### Overall Approach

Making Connections Across Ideas and Representations In advanced physics courses, we want students to build on a strong network of prior physics and math knowledge. Students often have trouble recruiting knowledge and problems they've solved from previous courses. Different students have different experiences in prerequisite courses, where some topics might not be covered, different notations are used, or it's just been a long time.

This course is structured to help students solidify and build on these previous experiences. Formative assessments - especially small whiteboard questions - are used to identify what students really know about a topic and what notation is familiar so that the instructor can use language and examples that students understand. The problems students solved in introductory physics - a plane pendulum, a block sliding down an incline plane, a train moving at constant speed, a projectile - become special cases to evaluate for more advanced problems. Kinematics graphs become spacetime diagrams. The unit circle becomes the unit hyperbola. Newton's 2nd Law becomes a differential equation to solve and a way of interpreting the Euler-Lagrange equation. Energy becomes a way to find equations of motion.

Evaluative Sensemaking In advanced theoretical courses, the problems have symbolic parameters and the algebra is often long and messy. In contrast, in introductory courses, students are given numbers, problems are made into numerous “different” problems by changing the parameters, and the algebra is relatively straightforward. In this class, the answers to problems are symbolic and the algebra more involved than in introductory physics.

To help students to deal with this, students are guided to build a habit of evaluating their own answers: Does their answer make sense? In order to do this, they have to know what they expect their answer to be like. Anticipating the features of the correct answer by making educated guesses is really hard for students at this level. During class, students work in groups to solve very challenging, symbolic problems. The in-class discussion of these problems emphasize articulating expectations at the beginning and then checking answers against those expectations. If the answer and expectation disagree, students must decide whether they need to refine their intuitions about their expectation or if their answer is in fact incorrect. Homework problems are structured so that early assignments have lots of supports for evaluative sensemaking. Over time, the supports fade while students understand that they are expected to develop a sensemaking habit. This process helps students become more sophisticated learners and problem solvers.

Mathematical & Notational Sophistication The calculations in this course are more advanced than in calculus-based introductory physics. Students solve problems without any numbers and need to grapple with the difference between constants, parameters, and variables. Students use matrix multiplication and hyperbolic trig to perform Lorentz transformations. I introduce the idea of “zapping with d” (exact differentials) for thinking about derivatives as ratios of small changes, and use this idea for deriving the Euler-Lagrange equation and for motivating separable ODE's. My students take partial derivatives with respect to variables that are not "x" or "t", and take total time derivatives of multivariable functions. I use dot notation for indicating time derivatives. Students use power series for considering limiting cases. They use curvilinear basis vectors to talk about directions that respect symmetries. An interesting synergy in this class is that hyperbolic trig functions show up in special relativity, a Lagrangian problem, and in quadratic drag problems.

Collaboration & Communication Working well and sharing ideas with others is crucial for doing physics. This class (with Contemporary Challenges) is the first class that is largely composed of physics majors. Some students took introductory physics at this university; others are transfer students. Many students hadn't decided to major in physics until recently and thought of themselves as engineers/chemists/mathematicians/etc while taking the introductory courses. The teaching strategies in this course are aimed at helping the students learn to do physics inclusively. Students see that making mistakes is common and important for learning physics and that classmates have valuable ideas and insights. Some students start the class believing that physics is about manipulating equations. Students are expected to explain their thinking with words, diagrams, graphs, and equations on homework and exams. During class, students are asked to share their reasoning with their small groups and the whole class. Class meetings feel more like discussions among students and instructors that adapt to the content of what people are saying rather than a performed presentation of lecture notes.

#### Content

Main Physics Topic Areas

• Special Relativity
• Velocity-Dependent Forces
• Introduction to Lagrangian & Hamiltonian Mechanics

Main Sensemaking Emphasis:

• Strategically Using“Weird” Coordinate Systems (Spacetime Coordinates, Curvilinear Coordinates, Generalized Coordinates)
• Coordinating Diagrams, Graphs, and Algebraic Equations
• Evaluating Ugly Algebraic Equations, especially: Checking and Balancing “Types of Beasts” (including dimensions of physical quantities), Evaluating Special Cases, the Examining the Behavior of Functions, Comparing Relativistic & Classical Contexts, Comparing Symbolic Quantities

Active Teaching Strategies:

• Small Whiteboard Questions w/ Peer Instruction
• Small Group Problems w/ Students Presenting Solutions
• Kinesthetic Activities & Visualization Props
• Computer Visualization

#### Level & Prerequisites

This middle division course aims to help students make the transition from the calculus-based introductory physics courses. The problems are longer, symbolic, and more abstract than students have previously done.

Prerequisite Knowledge:

Students are expected to have taken calculus-based introductory physics & multivariable calculus courses that discussed:

• Translational & rotational kinematics
• Projectile motion
• An object slides/rolling down an inclined plane
• Harmonic oscillator - mass on a spring and plane pendulum
• Newton's 2nd Law for linear and rotational motion
• Potential energy
• Translational & rotational kinetic energy
• Conservation of energy & momentum
• Partial derivatives
• Polar, cylindrical, & spherical coordinate systems