Preface

This document is a record for the paths that a junior and I as his mentor explored for Summer of Science. It may be used by anyone wanting to study classical field theory.

(The format is free for modification as you like and whatever works best for you.)

Chapter 1

Introduction to the new area and packing up for the journey.

Review

Week 1 May (6-11) -

  • Reading Marsden’s book, progress is slow. First chapter, continuity, to be more specific. Parallelly, also reading Jackson’s book. (finished the first chapter)

Week 2 May (13-20)

  • Now reading Marsden’s book, started the local differential calculus part. Everything now is just theorems so do these go into the final report? Everything you will ever study in life is just study of structures. Structures are defined by delineating their boundaries. Theorems are exceptionally well crafted boundaries= special structures. Yes, they form the basis of your report.

Week 3 May (21-27)

  • Videos

  • This is to make sure you get used to summarizing. Review doesn’t mean writing down everything you’ve covered but to condense it down to the key points enough to give an overview without giving too many details.)

Formalities : Midterm report

  • Jackson will do electrostatics for the 1st 4 chapters then magnetostatics (which is what you cover in PH108 with the shut up and calculate approach). This part does not need relativity as you can imagine relativity comes in only for high velocity, here its zero. Now E, B are fields that satisfy the wave equation, which travel at the speed of light so electrodynamics involving charges in motion necessarily needs relativity. In fact the Special theory of relativity is nothing but a theory of spacetime which preserves a symmetry in the Maxwell’s equations- namely Lorentz symmetry. Einstein’s Gravity (not Newton’s gravity) comes only after General theory of relativity, in turn for which you need to do the math first or you will not be able to appreciate it. That’s just the standard way theoretical physics curriculum is organized.

  • Historically people have gone from Electricity, Magnetism,Gravity>Newton’s LOM>Thermodynamics and Classical Mechanics> Electrodynamics (Maxwell) and Statistical Mechanics( Boltzmann)> Special Relativity and Quantum Mechanics> Quantum Statistical Mechanics>General Relativity (using sophisticated math machinery: Diff Geometry, Sub Riemannian Geometry)>Quantum Chromodynamics>Standard Model> Still looking for a Theory of Quantum Gravity

  • This above path is full of twists and turns so its not recommended that you go along that unless you want to be a historian of science or understand ways of development of scientific ideas/ structures. The straightforward path is the one I have mentioned below in the resources: Math background(=Multivar Calc+Manifolds+Tensors+Riemannian Geometry), classical mech(Calculus of variations + hamiltonian + lagrangian formulation + symmetry+systems with constraints), electrodynamics (which includes Special Relativity (SR) see later chapters of Griffiths or Jackson), Gravity (which includes General Relativity which is just SR generalized to curved spacetime= manifold which is not euclidean hence Riemannian Geometry needed).

  • Honestly speaking even if you learn everything that’s been developed till now, you are still not going to “see” or reach “gravity” in its ultimate form.

Resources (feel free to add your own) :

Mathematical background

  • V.I. Arnold methods of classical mechanics.
  • Abraham, Marsden foundations of mechanics.
  • Bugolubov N, Logunov A, Osak and Todorov Introduction to Axiomatic Field Theory.
  • Symplectic geometry of integrable Hamiltonian systems. Audin, Da Silva
  • Zehnder, Eduard- Lectures on dynamical systems: Hamiltonian vector fields and symplectic capacities

Classical mechanics

  • Goldstein classical mechanics
  • *Landau and Lifshcitz classical theory of fields
  • The L and L series is a very good standard if you’re looking to cover theoretical physics on your own.
  • Santilli Foundations of theoretical mechanics
  • Classical mechanics : systems of particles and Hamiltonian dynamics

Electromagnetic theory=( Electrostatics + Special Relativity + Electrodynamics)

  • Introduction to electrodynamics Griffiths
  • Classical electrodynamics Jackson
  • These two cover standard topics on electromagnetic fields. Chapters on fields in matter/ scattering/ special techniques to solve * Laplace equations can be skipped.

Gravity= General Relativity

  • Logunov Theory of Gravity

General Physics

  • R. P.Feynman, R.B. Leighton and M. Sands, Feynman Lectures-Vol 1: chap 26-34, Narosa Publishing House, 1988, New Delh
  • R.P.Feynman, R.B. Leighton and M. Sands, Feynman Lectures-Vol 2, Narosa Publishing House, 1988, New Delhi
  • E. M. Purcell, Berkeley Physics Course-Vol 2, Electricity and Magnetism, McGraw Hill Book Co., 1981
  • F. S. Crawford, Berkeley Physics Course-Vol 3, Waves, McGraw Hill Book Co., 1968