Here you can find a fair amount of worked problems and self-study material related to the theory of Lagrangians, Hamiltonians, and Noether’s theorem (475 pages.) 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.

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 git@github.com: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
  • PART 1: PHY354 (UOFT ADVANCED CLASSICAL MECHANICS) LECTURE NOTES
  • 1 Runge-Lenz vector conservation
  • 1.1 Motivation
  • 1.2 Motivation: The Kepler problem
  • 1.3 Runge-Lenz vector
  • 1.3.1 Verify the conservation assumption
  • 2 Phase Space and Trajectories
  • 2.1 Phase space and phase trajectories
  • 2.1.1 Applications of H
  • 2.1.2 Poisson brackets
  • 2.2 Adiabatic changes in phase space and conserved quantities
  • 2.3 Appendix I. Poisson brackets of angular momentum
  • 2.4 Appendix II. EOM for the variable length pendulum
  • 3 Rigid body motion
  • 3.1 Rigid body motion
  • 3.1.1 Setup
  • 3.1.2 Degrees of freedom
  • 3.2 Kinetic energy
  • 4 Classical Mechanics Euler Angles
  • 4.1 Pictorially
  • 4.2 Relating the two pairs of coordinate systems
  • 5 Parallel axis theorem
  • PART 2: WORKED PROBLEMS
  • 6 PHY354 Problem set 1 (2012)
  • 6.1 Problems
  • 6.2 Solutions
  • 7 PHY354 Problem set 2 (2012)
  • 7.1 Problems
  • 7.2 Solutions
  • 8 Attempts at solutions for some Goldstein Mechanics problems
  • 8.1 Problems
  • 8.2 Solutions
  • 9 Solutions to David Tong’s mf1 Lagrangian problems
  • 9.1 Problems
  • 9.2 Solutions
  • 10 Solutions to some Landau mechanics problems
  • 10.1 Problems
  • 10.2 Solutions
  • 11 Other worked problems
  • lagrangian topics
  • PART 3: NEWTON’S LAW FROM LAGRANGIAN
  • 12.1 The mistake hiding above
  • 12.2 Equations of motion for vectors in a general frame
  • 12.3 Appendix. Scratch calculations
  • 12.4 frame vector in terms of metric tensor, and reciprocal pairs
  • 12.5 Gradient calculation for an absolute vector magnitude function
  • 13 Covariant Lagrangian, and electrodynamic potential
  • 13.1 Motivation
  • 13.2 Guess at the Lagrange equations for relativistic correctness
  • 13.2.1 Try it with a non-velocity dependent potential
  • 13.2.2 Velocity dependent potential
  • 14 Vector canonical momentum
  • 15 Direct variation of Maxwell equations
  • 15.1 Motivation, definitions and setup
  • 15.1.1 Tensor form of the field
  • 15.1.2 Maxwell’s equation in tensor form
  • 15.2 Field square
  • 15.2.1 Scalar part
  • 15.2.2 Pseudoscalar part
  • 15.3 Variational background
  • 15.3.1 One dimensional purely kinetic Lagrangian
  • 15.3.2 Electrostatic potential Lagrangian
  • 15.4 Lagrangian generation of the vector part of Maxwell’s equation
  • 15.5 Lagrangian generation of the trivector (dual) part of Maxwell’s equation
  • 15.6 Complex Lagrangian used to generate the complete Maxwell equation
  • 15.7 Summary
  • 16 Revisit Lorentz force from Lagrangian
  • 16.1 Motivation
  • 16.2 Equations of motion
  • 16.3 Correspondence with classical form
  • 16.4 General potential
  • 17 Derivation of Euler-Lagrange field equations
  • 17.1 Motivation
  • 17.2 Deriving the field Lagrangian equations
  • 17.2.1 First order Taylor expansion of a multi variable function
  • 17.2.2 First order expansion of the Lagrangian function
  • 17.2.3 Example for clarification
  • 17.2.4 Calculation of the action for the general field Lagrangian
  • 17.3 Verifying the equations
  • 17.3.1 Maxwell’s equation derivation from action
  • 17.3.2 Electrodynamic Potential Wave Equation
  • 17.3.3 Schr\366dinger’s equation
  • 17.3.4 Relativistic Schr\366dinger’s equation
  • 18 Tensor Derivation of Covariant Lorentz Force from Lagrangian
  • 18.1 Motivation
  • 18.2 Calculation
  • 18.3 Compare for reference to GA form
  • 19 Euler Lagrange Equations
  • 19.1 Scalar form of Euler-Lagrange equations
  • 19.1.1 Some comparison to the Goldstein approach
  • 19.1.2 Noether’s theorem
  • 19.2 Vector formulation of Euler-Lagrange equations
  • 19.2.1 Simple case. Unforced purely kinetic Lagrangian
  • 19.2.2 Position and velocity gradients in the configuration space
  • 19.3 Example applications of Noether’s theorem
  • 19.3.1 Angular momentum in a radial potential
  • 19.3.2 Hamiltonian
  • 19.3.3 Covariant Lorentz force Lagrangian
  • 19.3.4 Vector Lorentz force Lagrangian
  • 19.3.5 An example where the transformation has to be evaluated at fixed point
  • 19.3.6 Comparison to cyclic coordinates
  • 19.4 Appendix
  • 19.4.1 Noether’s equation derivation, multivariable case
  • 20 Lorentz Invariance of Maxwell Lagrangian
  • 20.1 Working in multivector form
  • 20.1.1 Application of Lorentz boost to the field Lagrangian
  • 20.1.2 Lorentz boost applied to the Lorentz force Lagrangian
  • 20.2 Repeat in tensor form
  • 20.2.1 Translating versors to matrix form
  • 20.3 Translating versors tensor form
  • 20.3.1 Expressing vector Lorentz transform in tensor form
  • 20.3.2 Misc notes
  • 20.3.3 Expressing bivector Lorentz transform in tensor form
  • 21 Lorentz transform Noether current (interaction Lagrangian)
  • 21.1 Motivation
  • 21.2 Covariant result
  • 21.3 Expansion in observer frame
  • 21.4 In tensor form
  • 22 Field form of Noether’s Law
  • 22.1 Derivation
  • 22.2 Examples
  • 22.2.1 Klein-Gordan Lagrangian invariance under phase change
  • 22.2.2 Lorentz boost and rotation invariance of Maxwell Lagrangian
  • 22.2.3 Questions
  • 22.2.4 Expansion for x-axis boost
  • 22.2.5 Expansion for rotation or boost
  • 22.3 Multivariable derivation
  • 23 Lorentz force Lagrangian with conjugate momentum
  • 23.1 Motivation
  • 23.2 Lorentz force Lagrangian with conjugate momentum
  • 23.3 On terminology. The use of the term conjugate momentum
  • 24 Tensor derivation of Maxwell equation (non-dual part) from Lagrangian
  • 24.1 Motivation
  • 24.2 Lagrangian
  • 24.3 Calculation
  • 24.3.1 Preparation
  • 24.3.2 Derivatives
  • 24.3.3 Compare to STA form
  • 25 Canonical energy momentum tensor and Lagrangian translation
  • 25.1 Motivation and direction
  • 25.2 On translation and divergence symmetries
  • 25.2.1 Symmetry due to total derivative addition to the Lagrangian
  • 25.2.2 Some examples adding a divergence
  • 25.2.3 Symmetry for Wave equation under spacetime translation
  • 25.2.4 Symmetry condition for arbitrary linearized spacetime translation
  • 25.3 Noether current
  • 25.3.1 Vector parametrized Noether current
  • 25.3.2 Comment on the operator above
  • 25.3.3 In tensor form
  • 25.3.4 Multiple field variables
  • 25.3.5 Spatial Noether current
  • 25.4 Field Hamiltonian
  • 25.5 Wave equation
  • 25.5.1 Tensor components and energy term
  • 25.5.2 Conservation equations
  • 25.5.3 Invariant length
  • 25.5.4 Diagonal terms of the tensor
  • 25.5.5 Momentum
  • 25.6 Wave equation. GA form for the energy momentum tensor
  • 25.6.1 Calculate GA form
  • 25.6.2 Verify against tensor expression
  • 25.6.3 Invariant length
  • 25.6.4 Energy and Momentum split (again)
  • 25.7 Scalar Klein Gordon
  • 25.8 Complex Klein Gordon
  • 25.8.1 Tensor in GA form
  • 25.8.2 Tensor in index form
  • 25.8.3 Invariant Length?
  • 25.8.4 Divergence relation
  • 25.8.5 TODO
  • 25.9 Electrostatics Poisson Equation
  • 25.9.1 Lagrangian and spatial Noether current
  • 25.9.2 Energy momentum tensor
  • 25.9.3 Divergence and adjoint tensor
  • 25.10 Schr\366dinger equation
  • 25.11 Maxwell equation
  • 25.11.1 Lagrangian
  • 25.11.2 Energy momentum tensor
  • 25.11.3 Index form of tensor
  • 25.11.4 Expansion in terms of E and B
  • 25.11.5 Adjoint
  • 25.12 Nomenclature. Linearized spacetime translation
  • 26 Comparison of two covariant Lorentz force Lagrangians
  • 26.1 Motivation
  • 26.2 Lagrangian with Quadratic Velocity
  • 26.3 Lagrangian with Absolute Velocity
  • 27 Translation and rotation Noether field currents
  • 27.1 Motivation
  • 27.2 Field Euler-Lagrange equations
  • 27.3 Field Noether currents
  • 27.4 Spacetime translation symmetries and Noether currents
  • 27.4.1 On the symmetry
  • 27.4.2 Existence of a symmetry for translational variation
  • 27.4.3 Noether current derivation
  • 27.4.4 Relating the canonical energy momentum tensor to the Lagrangian gradient
  • 27.5 Noether current for incremental Lorentz transformation
  • 27.5.1 An example of the symmetry
  • 27.5.2 General existence of the rotational symmetry
  • 28 Lorentz force from Lagrangian (non-covariant)
  • 28.1 Motivation
  • 28.2 Guts
  • 29 Spherical polar pendulum for one and multiple masses, and multivector Euler-Lagrange formulation
  • 29.1 Motivation
  • 29.2 Kinetic energy for the single pendulum case
  • 29.3 Two and multi particle case
  • 29.4 Building up to the Multivector Euler-Lagrange equations
  • 29.4.1 A first example to build intuition
  • 29.4.2 A second example
  • 29.4.3 A third example
  • 29.5 Multivector Euler-Lagrange equations
  • 29.5.1 Derivation
  • 29.5.2 Work some examples
  • 29.6 Evaluating the pendulum Euler-Lagrange equations (scalar, bivector parametrized KE)
  • 29.7 Appendix calculation. A verification that the Kinetic matrix product is a real scalar
  • 30 Spherical polar pendulum for one and multiple masses (Take II)
  • 30.1 Motivation
  • 30.2 The Lagrangian
  • 30.3 Some tidy up
  • 30.4 Evaluating the Euler-Lagrange equations
  • 30.5 Hamiltonian form and linearization
  • 30.5.1 Thoughts about the Hamiltonian singularity
  • 30.6 A summary
  • PART 4: RANDOM INDEPENDENT STUDY NOTES
  • 31 Potential and Kinetic Energy
  • 31.1
  • 31.1.1 Work with a specific example. Newtonian gravitational force
  • 32 Compare some wave equation’s and their Lagrangians
  • 32.1 Motivation
  • 32.2 Vibrating object equations
  • 32.2.1 One dimensional wave equation
  • 32.2.2 Higher dimension wave equation
  • 32.3 Electrodynamics wave equation
  • 32.3.1 Comparing with complex (bivector) version of Maxwell Lagrangian
  • 32.4 Quantum Mechanics
  • 32.4.1 Non-relativistic case
  • 32.4.2 Relativistic case. Klein-Gordon
  • 32.4.3 Dirac wave equation
  • 32.5 Summary comparison of all the second order wave equations
  • 33 Short metric tensor explanation
  • 33.1
  • 34 Hamiltonian notes
  • 34.1 Motivation
  • 34.2 Hamiltonian as a conserved quantity
  • 34.3 Some syntactic sugar. In vector form
  • 34.4 The Hamiltonian principle
  • 34.5 Examples
  • 34.5.1 Force free motion
  • 34.5.2 Linear potential (surface gravitation)
  • 34.5.3 Harmonic oscillator (spring potential)
  • 34.5.4 Harmonic oscillator (change of variables.)
  • 34.5.5 Force free system dependent on only differences
  • 34.5.6 Gravitational potential
  • 34.5.7 Pendulum
  • 34.5.8 Spherical pendulum
  • 34.5.9 Double and multiple pendulums, and general quadratic velocity dependence
  • 34.5.10 Dangling mass connected by string to another
  • 34.5.11 Non-covariant Lorentz force
  • 34.5.12 Covariant force free case
  • 34.5.13 Covariant Lorentz force
  • 35 Linear transformations that retain two by two positive definiteness
  • 35.1 Motivation
  • 35.2 Guts
  • 36 Lagrangian and Euler-Lagrange equation evaluation for the spherical N-pendulum problem
  • 36.1 Introduction
  • 36.2 Diving right in
  • 36.2.1 Single spherical pendulum
  • 36.2.2 Spherical double pendulum
  • 36.2.3 N spherical pendulum
  • 36.3 Evaluating the Euler-Lagrange equations
  • 36.4 Summary
  • 36.5 Conclusions and followup
  • 37 1D forced harmonic oscillator. Quick solution of non-homogeneous problem
  • 37.1 Motivation
  • 37.2 Guts
  • 38 Integrating the equation of motion for a one dimensional problem
  • 38.1 Motivation
  • 38.2 Guts
  • 39 Notes on Goldstein’s Routh’s procedure
  • 39.1 Motivation
  • 39.2 Spherical pendulum example
  • 39.3 Simpler planar example
  • 39.4 Polar form example
  • 40 Hoop and spring oscillator problem
  • 40.1 Motivation
  • 40.2 Guts
  • 41 Attempts at calculating potential distribution for infinite homogeneous plane
  • 41.1 Motivation
  • 41.2 Forces and potential for an infinite homogeneous plane
  • 41.2.1 Calculating the potential from the force
  • 41.2.2 Calculating the potential directly
  • 42 Potential for an infinitesimal width infinite plane. Take III
  • 42.1 Document generation experiment
  • 42.2 Guts
  • 43 Potential due to cylindrical distribution
  • 43.1 Motivation
  • 43.2 Attempting to evaluate the integrals
  • 43.2.1 Alternate evaluation order?
  • PART 5: APPENDIX
  • 44 Mathematica notebooks
  • 45 Chronological Index
  • Copyright
  • Document Version
  • Dedication
  • Preface
  • Contents
  • List of Figures
  • PART 1: PHY354 (UOFT ADVANCED CLASSICAL MECHANICS) LECTURE NOTES
  • 1 Runge-Lenz vector conservation
  • 1.1 Motivation
  • 1.2 Motivation: The Kepler problem
  • 1.3 Runge-Lenz vector
  • 1.3.1 Verify the conservation assumption
  • 2 Phase Space and Trajectories
  • 2.1 Phase space and phase trajectories
  • 2.1.1 Applications of H
  • 2.1.2 Poisson brackets
  • 2.2 Adiabatic changes in phase space and conserved quantities
  • 2.3 Appendix I. Poisson brackets of angular momentum
  • 2.4 Appendix II. EOM for the variable length pendulum
  • 3 Rigid body motion
  • 3.1 Rigid body motion
  • 3.1.1 Setup
  • 3.1.2 Degrees of freedom
  • 3.2 Kinetic energy
  • 4 Classical Mechanics Euler Angles
  • 4.1 Pictorially
  • 4.2 Relating the two pairs of coordinate systems
  • 5 Parallel axis theorem
  • PART 2: WORKED PROBLEMS
  • 6 PHY354 Problem set 1 (2012)
  • 6.1 Problems
  • 6.2 Solutions
  • 7 PHY354 Problem set 2 (2012)
  • 7.1 Problems
  • 7.2 Solutions
  • 8 Attempts at solutions for some Goldstein Mechanics problems
  • 8.1 Problems
  • 8.2 Solutions
  • 9 Solutions to David Tong’s mf1 Lagrangian problems
  • 9.1 Problems
  • 9.2 Solutions
  • 10 Solutions to some Landau mechanics problems
  • 10.1 Problems
  • 10.2 Solutions
  • 11 Other worked problems
  • lagrangian topics
  • PART 3: NEWTON’S LAW FROM LAGRANGIAN
  • 12.1 The mistake hiding above
  • 12.2 Equations of motion for vectors in a general frame
  • 12.3 Appendix. Scratch calculations
  • 12.4 frame vector in terms of metric tensor, and reciprocal pairs
  • 12.5 Gradient calculation for an absolute vector magnitude function
  • 13 Covariant Lagrangian, and electrodynamic potential
  • 13.1 Motivation
  • 13.2 Guess at the Lagrange equations for relativistic correctness
  • 13.2.1 Try it with a non-velocity dependent potential
  • 13.2.2 Velocity dependent potential
  • 14 Vector canonical momentum
  • 15 Direct variation of Maxwell equations
  • 15.1 Motivation, definitions and setup
  • 15.1.1 Tensor form of the field
  • 15.1.2 Maxwell’s equation in tensor form
  • 15.2 Field square
  • 15.2.1 Scalar part
  • 15.2.2 Pseudoscalar part
  • 15.3 Variational background
  • 15.3.1 One dimensional purely kinetic Lagrangian
  • 15.3.2 Electrostatic potential Lagrangian
  • 15.4 Lagrangian generation of the vector part of Maxwell’s equation
  • 15.5 Lagrangian generation of the trivector (dual) part of Maxwell’s equation
  • 15.6 Complex Lagrangian used to generate the complete Maxwell equation
  • 15.7 Summary
  • 16 Revisit Lorentz force from Lagrangian
  • 16.1 Motivation
  • 16.2 Equations of motion
  • 16.3 Correspondence with classical form
  • 16.4 General potential
  • 17 Derivation of Euler-Lagrange field equations
  • 17.1 Motivation
  • 17.2 Deriving the field Lagrangian equations
  • 17.2.1 First order Taylor expansion of a multi variable function
  • 17.2.2 First order expansion of the Lagrangian function
  • 17.2.3 Example for clarification
  • 17.2.4 Calculation of the action for the general field Lagrangian
  • 17.3 Verifying the equations
  • 17.3.1 Maxwell’s equation derivation from action
  • 17.3.2 Electrodynamic Potential Wave Equation
  • 17.3.3 Schr\366dinger’s equation
  • 17.3.4 Relativistic Schr\366dinger’s equation
  • 18 Tensor Derivation of Covariant Lorentz Force from Lagrangian
  • 18.1 Motivation
  • 18.2 Calculation
  • 18.3 Compare for reference to GA form
  • 19 Euler Lagrange Equations
  • 19.1 Scalar form of Euler-Lagrange equations
  • 19.1.1 Some comparison to the Goldstein approach
  • 19.1.2 Noether’s theorem
  • 19.2 Vector formulation of Euler-Lagrange equations
  • 19.2.1 Simple case. Unforced purely kinetic Lagrangian
  • 19.2.2 Position and velocity gradients in the configuration space
  • 19.3 Example applications of Noether’s theorem
  • 19.3.1 Angular momentum in a radial potential
  • 19.3.2 Hamiltonian
  • 19.3.3 Covariant Lorentz force Lagrangian
  • 19.3.4 Vector Lorentz force Lagrangian
  • 19.3.5 An example where the transformation has to be evaluated at fixed point
  • 19.3.6 Comparison to cyclic coordinates
  • 19.4 Appendix
  • 19.4.1 Noether’s equation derivation, multivariable case
  • 20 Lorentz Invariance of Maxwell Lagrangian
  • 20.1 Working in multivector form
  • 20.1.1 Application of Lorentz boost to the field Lagrangian
  • 20.1.2 Lorentz boost applied to the Lorentz force Lagrangian
  • 20.2 Repeat in tensor form
  • 20.2.1 Translating versors to matrix form
  • 20.3 Translating versors tensor form
  • 20.3.1 Expressing vector Lorentz transform in tensor form
  • 20.3.2 Misc notes
  • 20.3.3 Expressing bivector Lorentz transform in tensor form
  • 21 Lorentz transform Noether current (interaction Lagrangian)
  • 21.1 Motivation
  • 21.2 Covariant result
  • 21.3 Expansion in observer frame
  • 21.4 In tensor form
  • 22 Field form of Noether’s Law
  • 22.1 Derivation
  • 22.2 Examples
  • 22.2.1 Klein-Gordan Lagrangian invariance under phase change
  • 22.2.2 Lorentz boost and rotation invariance of Maxwell Lagrangian
  • 22.2.3 Questions
  • 22.2.4 Expansion for x-axis boost
  • 22.2.5 Expansion for rotation or boost
  • 22.3 Multivariable derivation
  • 23 Lorentz force Lagrangian with conjugate momentum
  • 23.1 Motivation
  • 23.2 Lorentz force Lagrangian with conjugate momentum
  • 23.3 On terminology. The use of the term conjugate momentum
  • 24 Tensor derivation of Maxwell equation (non-dual part) from Lagrangian
  • 24.1 Motivation
  • 24.2 Lagrangian
  • 24.3 Calculation
  • 24.3.1 Preparation
  • 24.3.2 Derivatives
  • 24.3.3 Compare to STA form
  • 25 Canonical energy momentum tensor and Lagrangian translation
  • 25.1 Motivation and direction
  • 25.2 On translation and divergence symmetries
  • 25.2.1 Symmetry due to total derivative addition to the Lagrangian
  • 25.2.2 Some examples adding a divergence
  • 25.2.3 Symmetry for Wave equation under spacetime translation
  • 25.2.4 Symmetry condition for arbitrary linearized spacetime translation
  • 25.3 Noether current
  • 25.3.1 Vector parametrized Noether current
  • 25.3.2 Comment on the operator above
  • 25.3.3 In tensor form
  • 25.3.4 Multiple field variables
  • 25.3.5 Spatial Noether current
  • 25.4 Field Hamiltonian
  • 25.5 Wave equation
  • 25.5.1 Tensor components and energy term
  • 25.5.2 Conservation equations
  • 25.5.3 Invariant length
  • 25.5.4 Diagonal terms of the tensor
  • 25.5.5 Momentum
  • 25.6 Wave equation. GA form for the energy momentum tensor
  • 25.6.1 Calculate GA form
  • 25.6.2 Verify against tensor expression
  • 25.6.3 Invariant length
  • 25.6.4 Energy and Momentum split (again)
  • 25.7 Scalar Klein Gordon
  • 25.8 Complex Klein Gordon
  • 25.8.1 Tensor in GA form
  • 25.8.2 Tensor in index form
  • 25.8.3 Invariant Length?
  • 25.8.4 Divergence relation
  • 25.8.5 TODO
  • 25.9 Electrostatics Poisson Equation
  • 25.9.1 Lagrangian and spatial Noether current
  • 25.9.2 Energy momentum tensor
  • 25.9.3 Divergence and adjoint tensor
  • 25.10 Schr\366dinger equation
  • 25.11 Maxwell equation
  • 25.11.1 Lagrangian
  • 25.11.2 Energy momentum tensor
  • 25.11.3 Index form of tensor
  • 25.11.4 Expansion in terms of E and B
  • 25.11.5 Adjoint
  • 25.12 Nomenclature. Linearized spacetime translation
  • 26 Comparison of two covariant Lorentz force Lagrangians
  • 26.1 Motivation
  • 26.2 Lagrangian with Quadratic Velocity
  • 26.3 Lagrangian with Absolute Velocity
  • 27 Translation and rotation Noether field currents
  • 27.1 Motivation
  • 27.2 Field Euler-Lagrange equations
  • 27.3 Field Noether currents
  • 27.4 Spacetime translation symmetries and Noether currents
  • 27.4.1 On the symmetry
  • 27.4.2 Existence of a symmetry for translational variation
  • 27.4.3 Noether current derivation
  • 27.4.4 Relating the canonical energy momentum tensor to the Lagrangian gradient
  • 27.5 Noether current for incremental Lorentz transformation
  • 27.5.1 An example of the symmetry
  • 27.5.2 General existence of the rotational symmetry
  • 28 Lorentz force from Lagrangian (non-covariant)
  • 28.1 Motivation
  • 28.2 Guts
  • 29 Spherical polar pendulum for one and multiple masses, and multivector Euler-Lagrange formulation
  • 29.1 Motivation
  • 29.2 Kinetic energy for the single pendulum case
  • 29.3 Two and multi particle case
  • 29.4 Building up to the Multivector Euler-Lagrange equations
  • 29.4.1 A first example to build intuition
  • 29.4.2 A second example
  • 29.4.3 A third example
  • 29.5 Multivector Euler-Lagrange equations
  • 29.5.1 Derivation
  • 29.5.2 Work some examples
  • 29.6 Evaluating the pendulum Euler-Lagrange equations (scalar, bivector parametrized KE)
  • 29.7 Appendix calculation. A verification that the Kinetic matrix product is a real scalar
  • 30 Spherical polar pendulum for one and multiple masses (Take II)
  • 30.1 Motivation
  • 30.2 The Lagrangian
  • 30.3 Some tidy up
  • 30.4 Evaluating the Euler-Lagrange equations
  • 30.5 Hamiltonian form and linearization
  • 30.5.1 Thoughts about the Hamiltonian singularity
  • 30.6 A summary
  • PART 4: RANDOM INDEPENDENT STUDY NOTES
  • 31 Potential and Kinetic Energy
  • 31.1
  • 31.1.1 Work with a specific example. Newtonian gravitational force
  • 32 Compare some wave equation’s and their Lagrangians
  • 32.1 Motivation
  • 32.2 Vibrating object equations
  • 32.2.1 One dimensional wave equation
  • 32.2.2 Higher dimension wave equation
  • 32.3 Electrodynamics wave equation
  • 32.3.1 Comparing with complex (bivector) version of Maxwell Lagrangian
  • 32.4 Quantum Mechanics
  • 32.4.1 Non-relativistic case
  • 32.4.2 Relativistic case. Klein-Gordon
  • 32.4.3 Dirac wave equation
  • 32.5 Summary comparison of all the second order wave equations
  • 33 Short metric tensor explanation
  • 33.1
  • 34 Hamiltonian notes
  • 34.1 Motivation
  • 34.2 Hamiltonian as a conserved quantity
  • 34.3 Some syntactic sugar. In vector form
  • 34.4 The Hamiltonian principle
  • 34.5 Examples
  • 34.5.1 Force free motion
  • 34.5.2 Linear potential (surface gravitation)
  • 34.5.3 Harmonic oscillator (spring potential)
  • 34.5.4 Harmonic oscillator (change of variables.)
  • 34.5.5 Force free system dependent on only differences
  • 34.5.6 Gravitational potential
  • 34.5.7 Pendulum
  • 34.5.8 Spherical pendulum
  • 34.5.9 Double and multiple pendulums, and general quadratic velocity dependence
  • 34.5.10 Dangling mass connected by string to another
  • 34.5.11 Non-covariant Lorentz force
  • 34.5.12 Covariant force free case
  • 34.5.13 Covariant Lorentz force
  • 35 Linear transformations that retain two by two positive definiteness
  • 35.1 Motivation
  • 35.2 Guts
  • 36 Lagrangian and Euler-Lagrange equation evaluation for the spherical N-pendulum problem
  • 36.1 Introduction
  • 36.2 Diving right in
  • 36.2.1 Single spherical pendulum
  • 36.2.2 Spherical double pendulum
  • 36.2.3 N spherical pendulum
  • 36.3 Evaluating the Euler-Lagrange equations
  • 36.4 Summary
  • 36.5 Conclusions and followup
  • 37 1D forced harmonic oscillator. Quick solution of non-homogeneous problem
  • 37.1 Motivation
  • 37.2 Guts
  • 38 Integrating the equation of motion for a one dimensional problem
  • 38.1 Motivation
  • 38.2 Guts
  • 39 Notes on Goldstein’s Routh’s procedure
  • 39.1 Motivation
  • 39.2 Spherical pendulum example
  • 39.3 Simpler planar example
  • 39.4 Polar form example
  • 40 Hoop and spring oscillator problem
  • 40.1 Motivation
  • 40.2 Guts
  • 41 Attempts at calculating potential distribution for infinite homogeneous plane
  • 41.1 Motivation
  • 41.2 Forces and potential for an infinite homogeneous plane
  • 41.2.1 Calculating the potential from the force
  • 41.2.2 Calculating the potential directly
  • 42 Potential for an infinitesimal width infinite plane. Take III
  • 42.1 Document generation experiment
  • 42.2 Guts
  • 43 Potential due to cylindrical distribution
  • 43.1 Motivation
  • 43.2 Attempting to evaluate the integrals
  • 43.2.1 Alternate evaluation order?
  • PART 5: APPENDIX
  • 44 Mathematica notebooks
  • 45 Chronological Index
  • Bibliography