Preface

Introduction to the course

For math concentrators, at some point (late into high school or in college) there is a change of perspective from computation to proof-based math. For example, one stops viewing matrices as large blobs of numbers but as coordinate representations of abstract linear transformations.

There is a similar shift in perspective in physics, when the emphasis changes from solving equations of motion to understanding the fundamental reasons they’re there in the first place. Key to navigating this change is understanding the role of symmetry and conservation laws, as well as the importance of operators in physics.

The focus of this course is not quantum theory (maybe at the end, time permitting), but the classical treatment of symmetry and conservation will help motivate much of quantum theory’s constructions.

Using these notes

These notes accompany the Fall 2024 iteration of Arthur Jaffe’s Mechanics course at Harvard (Physics 151). They recount main results and, occasionally, supplement Prof. Jaffe’s lecture notes on canvas. It can serve as a concise reminder of the results in lecture.

The main deliverables are:

  1. Covariant and extremal-action perspectives on Lagrangian mechanics.
  2. Connection between Lagrangian and Hamiltonian mechanics via the Legendre transform.
  3. Nother’s theorem: continuous symmetry \(\implies\) conserved quantity.
  4. Spacetime geometry: unitary representation of (a component of) the Lorentz group.

A note on notation

It is always important to be careful about derivative maps, where overloaded notation can frequently lead to confusion. We use the following standard notation for partial derivatives:
\[ \partial_{x_1} f(x_1, x_2) = \dfrac{\partial}{\partial x_1} f(x_1, x_2) \] For second-order derivatives, we adopt \[ \partial_{x_1, x_2}^2 f(x_1, x_2) = \dfrac{\partial^2}{\partial x_1\partial x_2} f(x_1, x_2) \] We also use the following not-so-standard notation for total derivatives: \[ d_t f(x_1(t), x_2(t)) = \dfrac d {dt} f(x_1(t), x_2(t)) \] When we wish to specify the point at which a derivative is evaluated, we write e.g.  \[ d_t\big|_0 f(x_1(t), x_2(t)) = x'_1(0) \partial_{x_1} f + x'_2(0) \partial_{x_2} f \]