Weekly Course Materials#
What This Course Is Really About#
Classical Mechanics is the physics of large, slow, mechanical systems—the world we see around us. But more importantly, it’s about learning to think like a physicist: observing nature, building mathematical models, and using those models to predict and explain how systems evolve.
Throughout this course, you’ll develop a toolkit for understanding motion, forces, and the deep principles that govern how nature changes. You’ll move from solving simple differential equations to analyzing chaotic systems and discovering that the laws of physics themselves can be expressed through variational principles—paths that minimize (or maximize) a particular quantity called the action.
This is a course about both computation and understanding. We’ll solve problems analytically when possible, numerically when necessary, and geometrically when it provides insight.
How the Course is Organized#
🎯 Part I: Foundations & Mathematical Modeling (Weeks 1–3)#
Big Question: How do we turn observations of nature into equations that let us predict the future?
You’ll start by learning how physicists build models. We begin with Newton’s Laws—which you’ve likely seen before—but we’ll use them to develop a systematic process: identify interactions, sketch the system, choose coordinates, apply the framework, and derive equations of motion.
What you’ll learn:
What it means to “model” a physical system
Free body diagrams and Newton’s Second Law in multiple dimensions
How to set up differential equations for falling objects, springs, and more
The role of computing in solving equations we can’t tackle by hand
Real-world connection: The process you learn here is used everywhere—from designing bridges to predicting planetary orbits to modeling epidemics.
⚙️ Part II: Tools & Conservation Laws (Weeks 4–6)#
Big Question: What quantities never change, and why?
Newton’s Second Law gives us differential equations, but solving them requires techniques and insight. You’ll learn how to leverage the conservation of energy and conservation of momentum—universal principles that apply to every physical system. These principles often let us understand a system’s behavior without solving complicated equations.
What you’ll learn:
The Work-Energy Theorem and conservation of energy
Linear and angular momentum conservation
Why conservative forces allow us to define potential energy
How to use conservation principles to solve problems elegantly
Real-world connection: Collisions, explosions, orbital mechanics—anywhere energy or momentum is conserved.
🌀 Part III: Nonlinear Dynamics & Oscillations (Weeks 7–10)#
Big Question: What happens when we can’t solve the equation analytically?
Most real systems are nonlinear—their behavior can’t be solved with a pencil and paper. Enter phase space: a way of visualizing how a system evolves without solving the differential equation. You’ll learn to read phase portraits like maps of a system’s destiny.
We’ll study oscillators in detail—from simple harmonic motion (which you can solve) to driven oscillators (where resonance can build dramatic amplitudes) to nonlinear systems where chaos lurks. You’ll discover that small changes in initial conditions can lead to wildly different outcomes—a signature of chaos.
What you’ll learn:
Phase space analysis and phase portraits
Critical points and stability
Damping and driving forces
Resonance and why it matters (bridges, radios, atoms)
Introduction to nonlinear dynamics and chaos
Real-world connection: The Tacoma Narrows Bridge collapse, tuning a radio, atomic absorption, weather prediction—all involve oscillations and resonance.
📐 Part IV: Variational Principles & Lagrangian Mechanics (Weeks 11–13)#
Big Question: Is there a deeper principle underlying Newton’s Laws?
There is. Nature follows paths that extremize a quantity called the action. This is the Calculus of Variations—a powerful mathematical framework that lets us derive equations of motion from an optimization principle rather than from forces.
You’ll solve classic problems (shortest path, Snell’s Law, the Brachistochrone) and discover the Lagrangian formulation of mechanics—an elegant reformulation of classical mechanics that connects to quantum mechanics and modern physics.
What you’ll learn:
The Calculus of Variations and the Euler-Lagrange equation
How to solve optimization problems in physics
The Lagrangian and Hamiltonian formulations of mechanics
Why these principles underlie all of physics
Real-world connection: These principles extend to quantum mechanics, field theory, and general relativity. Understanding them here prepares you for advanced physics.
Skills You’ll Develop#
By Week 3#
✓ Construct equations of motion from physical observations
✓ Solve differential equations using separation of variables
✓ Use numerical methods (Euler-Cromer) to simulate systems
By Week 6#
✓ Analyze systems using energy and momentum conservation
✓ Classify forces as conservative or non-conservative
✓ Solve complex dynamics problems without explicit solutions
By Week 10#
✓ Create and interpret phase portraits
✓ Classify critical points and determine stability
✓ Understand nonlinear behavior and sensitivity to initial conditions
By Week 13#
✓ Apply variational principles to derive equations of motion
✓ Solve optimization problems in physics
✓ Understand the deep mathematical structure of classical mechanics
Key Themes Throughout the Course#
1. Modeling: From Observations to Equations#
Physics is about building mathematical representations of the natural world. Each week, you’ll learn to identify simplifying assumptions, choose appropriate frameworks, and develop equations that capture essential physics.
2. Multiple Solution Methods#
The same problem can be solved in different ways—analytically, numerically, or geometrically (via phase space). Learning when to use each approach is part of becoming a physicist.
3. Conservation Laws#
Energy, momentum, and angular momentum are conserved. These principles transcend the specific details of a system and apply universally.
4. Nonlinearity is Ubiquitous#
Most real systems are nonlinear. Linear equations are the exception, and we solve them first because they’re tractable. But the world is full of nonlinear systems—and you’ll develop tools to understand them.
5. Computation is Essential#
Solving differential equations by hand is possible only for special cases. Computers let us explore systems we can’t solve analytically. You’ll learn to implement numerical methods and interpret the results.
Course Philosophy#
Physics is a Human Endeavor
Physics is a global activity conducted by humans across all cultures and throughout history. The examples you’ll study come from many traditions and places—ancient astronomy, Islamic mathematics, Chinese engineering, African and Indigenous astronomy, and modern physics across the world.
We acknowledge that physics as an institution has often erased the contributions of women, people of color, and early-career researchers. For example, the Euler-Cromer integration method was discovered by a graduate student named Abby Aspel but is rarely credited to her. As you learn these powerful tools and techniques, remember the humans behind them—and commit to building a more inclusive physics community.
Numerical Methods Are Real Physics
In this course, you’ll use computers to solve differential equations. Some instructors dismiss numerical solutions as “not real physics”—but this is wrong. Most systems in nature and industry are solved numerically. Learning to implement algorithms, interpret results, and understand sources of error is essential for modern physics.
The key principle: analytical solutions provide benchmarks to test numerical methods. We’ll solve some problems both ways so you can compare and build confidence.
How to Use These Materials#
Each week has a dedicated lecture note file with learning goals, conceptual foundations, worked examples, and applications.
Materials are not a substitute for textbooks—they’re adapted from https://mhjensen.github.io/Physics321/doc/web/course.html and inspired by the recommended textbooks in the course.
Come to class prepared. Read the notes beforehand and try the examples.
Combine with homework and projects. Lectures and notes develop concepts; problem sets and projects develop mastery.
Use multiple resources. Consult recommended textbooks, office hours, and your classmates.