In my (VERY ROUGH) classical mechanics notes you can find a fair amount of worked problems and self-study material related to the theory of Lagrangians, Hamiltonians, and Noether’s theorem. Also included are notes and a couple problems from occasional Wednesday audits of the 2012 session of PHY354H1S, Advanced Classical Mechanics, taught by Prof. Erich Poppitz, at the University of Toronto.

These notes are not self contained. In particular, there is fairly heavy use of geometric algebra in many of the problems, with assumptions that the reader is proficient with that algebra.

These notes (349 pages, 6″x9″) are available in the following formats:

- For free in PDF format (colour),
- an e-book from leanpub for $5+ (set your own price, and automatic updates). There is also a discussion forum available for purchasers,
- as latex sources (see below.)

There is very little structure in these notes. An extremely vicious edit is required, and it would be good to have standalone introductions to all the key ideas before diving into the problems.

I had a version of these notes available earlier in paperback on amazon, but now that I’ve ordered a copy for myself for editing purposes, I’ve unpublished it. Until I do some drastic editing, I am no longer going to make a paper version available for purchase, and it will show as “out of print“.

## Contributing.

Should you wish to actively contribute typo fixes (or additions, editing, …) to this book, you can do so by contacting me, or by forking your own copy of the associated git repositories and building the book pdf from source, and submitting a subsequent merge request.

git clone [email protected]:peeterjoot/latex-notes-compilations.git peeterjoot cd peeterjoot for i in figures/classicalmechanics classicalmechanics mathematica latex ; do git submodule update --init $i (cd $i ; git checkout master) done export PATH=`pwd`/latex/bin:$PATH cd classicalmechanics make

I reserve the right to impose dictatorial control over any editing and content decisions, and may not accept merge requests as-is, or at all. That said, I’ll probably not refuse reasonable suggestions or merge requests.

## Contents:

- Copyright
- Document Version
- Dedication
- Preface
- Contents
- List of Figures
- 1 Potential and Kinetic energy.
- 1.1 Potential and Kinetic Energy.
- 1.1.1 Work with a specific example. Newtonian gravitational force.
- 2 Calculus of variations.
- 2.1 Solutions.
- 3 Special relativity.
- 4 Action and Euler-Langrange equations.
- 4.1 Scalar form of Euler-Lagrange equations.
- 4.1.1 Some comparison to the Goldstein approach.
- 4.2 Problems.
- 4.3 Solutions.
- 5 Constraints.
- 5.1 Solutions.
- 6 Geometric algebra.
- 6.1 Euclidean geometric algebra.
- 6.2 Space Time Algebra (STA.)
- 6.2.1 Conventions.
- 6.2.2 Space Time Algebra (STA.)
- 6.3 Solutions.
- 7 Relativistic action.
- 7.1 In this chapter.
- 7.2 Euler-Lagrange equations.
- 7.2.1 Solutions.
- 7.3 Lorentz force equation.
- 7.4 Solutions.
- 8 Noether’s theorem.
- 8.0.1 Noether’s theorem.
- 8.1 Vector formulation of Euler-Lagrange equations.
- 8.1.1 Simple case. Unforced purely kinetic Lagrangian.
- 8.1.2 Position and velocity gradients in the configuration space.
- 8.2 Example applications of Noether’s theorem.
- 8.2.1 Angular momentum in a radial potential.
- 8.2.2 Hamiltonian.
- 8.2.3 Covariant Lorentz force Lagrangian.
- 8.3 Appendix.
- 8.3.1 Noether’s equation derivation, multivariable case.
- 9 Hamiltonian mechanics.
- 9.1 Motivation.
- 9.2 Hamiltonian as a conserved quantity.
- 9.3 Some syntactic sugar. In vector form.
- 9.4 The Hamiltonian principle.
- 9.5 Examples.
- 9.5.1 Force free motion.
- 9.5.2 Linear potential (surface gravitation).
- 9.5.3 Harmonic oscillator (spring potential).
- 9.5.4 Harmonic oscillator (change of variables.)
- 9.5.5 Force free system dependent on only differences.
- 9.5.6 Gravitational potential.
- 9.5.7 Pendulum.
- 9.5.8 Spherical pendulum.
- 9.5.9 Double and multiple pendulums, and general quadratic velocity dependence.
- 9.5.10 Non-covariant Lorentz force.
- 9.6 Solutions.
- 10 Routhian procedure.
- 10.1 Motivation.
- 10.2 Spherical pendulum example.
- 10.3 Simpler planar example.
- 10.4 Polar form example.
- 11 Rigid body motion.
- 11.1 Rigid body motion.
- 11.1.1 Setup.
- 11.1.2 Degrees of freedom.
- 11.2 Kinetic energy.
- 12 Euler angles.
- 12.1 Pictorially.
- 12.2 Relating the two pairs of coordinate systems.
- 13 Parallel axis theorem.
- 14 Phase Space and Trajectories.
- 14.1 Phase space and phase trajectories.
- 14.1.1 Applications of H
- 14.1.2 Poisson brackets.
- 14.2 Adiabatic changes in phase space and conserved quantities.
- 14.3 Appendix I. Poisson brackets of angular momentum.
- 14.4 Appendix II. EOM for the variable length pendulum.
- 15 Conserved quantities.
- 15.1 Runge-Lenz vector conservation.
- 15.1.1 Motivation.
- 15.1.2 Motivation: The Kepler problem.
- 15.1.3 Runge-Lenz vector.
- 15.2 Solutions.
- 16 Field Lagrangians.
- 16.0.1 Motivation.
- 16.0.2 Deriving the field Lagrangian equations.
- 16.0.3 Verifying the equations.
- 16.1 Solutions.
- 17 Noether’s theorem for fields.
- 17.1 Noether’s theorem.
- 17.1.1 Derivation.
- 17.1.2 Examples.
- 17.1.3 Multivariable derivation.
- 17.2 Translation and rotation Noether field currents.
- 17.2.1 Motivation.
- 17.2.2 Field Euler-Lagrange equations.
- 17.2.3 Field Noether currents.
- 17.2.4 Spacetime translation symmetries and Noether currents.
- 17.2.5 Noether current, infinitesimal Lorentz transformation.
- 17.3 Solutions.
- A Mathematica notebooks.
- B Spherical N-pendulum problem.
- B.1 Introduction.
- B.2 Diving right in.
- B.2.1 Single spherical pendulum.
- B.2.2 Spherical double pendulum.
- B.2.3 N spherical pendulum.
- B.3 Evaluating the Euler-Lagrange equations.
- B.4 Summary.
- C Vector canonical momentum.
- D Direct variation of Maxwell equations.
- D.1 Motivation, definitions and setup.
- D.1.1 Tensor form of the field.
- D.1.2 Maxwell’s equation in tensor form.
- D.2 Field square.
- D.2.1 Scalar part.
- D.2.2 Pseudoscalar part.
- D.3 Variational background.
- D.3.1 One dimensional purely kinetic Lagrangian.
- D.3.2 Electrostatic potential Lagrangian.
- D.4 Vector part of Maxwell’s equation.
- D.5 Trivector (dual) part of Maxwell’s equation.
- D.6 Maxwell equation with complex Lagrangian.
- D.7 Summary.
- E Lorentz Invariance of Maxwell Lagrangian.
- E.1 Working in multivector form.
- E.1.1 Application of Lorentz boost to the field Lagrangian.
- E.1.2 Lorentz boost applied to the Lorentz force Lagrangian.
- E.2 Repeat in tensor form.
- E.2.1 Translating versors to matrix form.
- E.3 Translating versors tensor form.
- E.3.1 Expressing vector Lorentz transform in tensor form.
- E.3.2 Misc notes.
- E.3.3 Expressing bivector Lorentz transform in tensor form.
- F Lorentz transform Noether current (interaction Lagrangian).
- F.1 Motivation.
- F.2 Covariant result.
- F.3 Expansion in observer frame.
- F.4 In tensor form.
- G Tensor Maxwell equation.
- G.1 Motivation.
- G.2 Lagrangian.
- G.3 Calculation.
- G.3.1 Preparation.
- G.3.2 Derivatives.
- G.3.3 Compare to STA form.
- H Canonical energy momentum tensor and translation.
- H.1 Motivation and direction.
- H.2 On translation and divergence symmetries.
- H.2.1 Symmetry due to total derivative addition to the Lagrangian.
- H.2.2 Some examples adding a divergence.
- H.2.3 Symmetry for Wave equation under spacetime translation.
- H.2.4 Symmetry condition for arbitrary linearized spacetime translation.
- H.3 Noether current.
- H.3.1 Vector parametrized Noether current.
- H.3.2 Comment on the operator above.
- H.3.3 In tensor form.
- H.3.4 Multiple field variables.
- H.3.5 Spatial Noether current.
- H.4 Field Hamiltonian.
- H.5 Wave equation.
- H.5.1 Tensor components and energy term.
- H.5.2 Conservation equations.
- H.5.3 Invariant length.
- H.5.4 Diagonal terms of the tensor.
- H.5.5 Momentum.
- H.6 Wave equation. GA form for the energy momentum tensor.
- H.6.1 Calculate GA form.
- H.6.2 Verify against tensor expression.
- H.6.3 Invariant length.
- H.6.4 Energy and Momentum split (again).
- H.7 Scalar Klein Gordon.
- H.8 Complex Klein Gordon.
- H.8.1 Tensor in GA form.
- H.8.2 Tensor in index form.
- H.8.3 Invariant Length?
- H.8.4 Divergence relation.
- H.8.5 TODO.
- H.9 Electrostatics Poisson Equation.
- H.9.1 Lagrangian and spatial Noether current.
- H.9.2 Energy momentum tensor.
- H.9.3 Divergence and adjoint tensor.
- H.10 Schroedinger equation
- H.11 Maxwell equation.
- H.11.1 Lagrangian.
- H.11.2 Energy momentum tensor.
- H.11.3 Index form of tensor.
- H.11.4 Adjoint.
- H.12 Nomenclature. Linearized spacetime translation.
- Index
- Bibliography

## Changelog:

classicalmechanics.V0.1.15-3 Dec 31 2020, (commit 1a33e9530f5)

- Switch to plain L for Lagrangians and calL for Lagrangian densities (use \Lq macro to allow for easy switch back.)
- New STA content (rough): reciprocal frames, gradient, vector derivative, line and surface integrals and fundamental theorems for both.

classicalmechanics.V0.1.13-5.pdf, Oct 24 2020, (commit 7b9e0539da)

Pruned 87 pages (some just tossed, other stuff rewritten.)

- junk: 1dharmonicOsc.tex 1dpotentialIntegral.tex InfinitePlanePotentialTakeIII.tex cylinderPotential.tex infiniteSheetPotentials.tex lForceLag2.tex lorentzForce.tex lorentzForcePQA.tex lorentzMetricTensor.tex maxwellTensorLagrangian.tex multiPendulumSpherical2.tex newtonianLagrangianAndGradient.tex quadraticForm.tex sPolarMultiPendulum.tex srLagrangian.tex waveLagrangian.tex
- new chapter content: covariantSTA.tex, sta.tex (replaces various poorer versions)
- convert jackson12Dash9 into problem, and fix gutter issues.
- move remainder of \part{Lagrangian Topics} to appendix.
- junk a bunch of stuff
- move multiPendulumSphericalMatrix.tex
- split galagrangian/eulerLagrange.tex into E-L and Noether parts.
- split off waveLagrangianDirac.tex, waveLagrangianKG.tex, waveLagrangianQM.tex
- move some of mine/waveLagrangian.tex into problems.
- convert hoopSpring.tex into a problem.
- Try to impose some basic structure.

classicalmechanics.V0.1.12-22.pdf, Oct 12 2020, (commit 75aaee7e385)

This was the first edition made available on KDP (436 pages).

- Converted dmath* and equation* s to labelled equations.
- Removed sentances that started with “Or,”.
- Internal cleanups: remove unless {aligned} enviornments, subequations of one equation, switch from {align} envionrment to equation + aligned.
- Fixup equation grammar in many places (periods and commas)
- Removed lots of the blank lines that forced undesirable latex paragraph breaks.
- Lots of gutter and equation overflow fixes (many caused by the 6×9 transition.)
- Remove parts.
- Updated email address.
- Listing hyphenation fix.
- url fixes.
- thethe.py fixes.
- Switch to 6×9 format
- Move chapter headings into chapters.tex