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Archive for the ‘geometric algebra’ Category

new post of geometric algebra notes, and first post of classicalmechanics notes.

Posted by peeterjoot on January 6, 2013

I’ve posted a new version of my Geometric algebra notes. The changelog (below) for this version is minimal (previous changelog).

There is, however, a significant change in this version of these notes.  I’ve moved a huge chunk out (most of the Lagrangian and Hamiltonian stuff) into a separate pdf on classical mechanics.

Also included in these classical mechanics notes is a few lecture notes from phy354 (UofT Advanced classical mechanics, taught by Prof Erich Poppitz).  I’d audited a few of those lectures, and did the first problem set (and subset of the second).

gabook changelog

January 04, 2013 Tangent planes and normals in three and four dimensions
Figure out how to express a surface normal in 3d and a “volume” normal in 4d.

Sept 2, 2012 Plane wave solutions in linear isotropic charge free media using Geometric Algebra
Work through the plane wave solution to Maxwell’s equation in linear isotropic charge free media without boundary value constraints. I may have attempted to blunder through this before, but believe this to be more clear than any previous attempts. What’s missing is relating this to polarization states of different types and relationships to Jones vectors and so forth. Also, it’s likely possible to express things in a way that doesn’t require taking any real parts provided one uses the pseudoscalar instead of the scalar complex imaginary appropriately.

Mar 16, 2012 Geometric Algebra. The very quickest introduction.

Jan 27, 2012 Infinitesimal rotations.
Derive the cross product result for infinitesimal rotations with and without GA.

Classical mechanics changelog

January 06, 2013 Parallel axis theorem
class notes from course audit

January 05, 2013 Problem set 2 (2012)
incomplete attempt at the problem set 2 questions.

December 27, 2012 Dipole Moment induced by a constant electric field

Jul 14, 2012 Some notes on a Landau mechanics problem

Mar 21, 2012 Classical Mechanics Euler Angles

Mar 7, 2012 Rigid body motion.

Feb 29, 2012 Phase Space and Trajectories.

Feb 27, 2012 Potential due to cylindrical distribution.

Feb 24, 2012 Potential for an infinitesimal width infinite plane. Take III

Feb 19, 2012 Attempts at calculating potential distribution for infinite homogeneous plane.

Feb 11, 2012 Runge-Lenz vector conservation
phy354 lecture notes on the Runge-Lenz vector and its use in the Kepler problem.

Jan 24, 2012 PHY354 Advanced Classical Mechanics. Problem set 1.

June 19, 2010 Hoop and spring oscillator problem.
A linear approximation to a hoop and spring problem.

Mar 3, 2010 Notes on Goldstein’s Routh’s procedure.
Puzzle through Routh’s procedure as outlined in Goldstein.

Feb 19, 2010 1D forced harmonic oscillator. Quick solution of non-homogeneous problem.
Solve the one dimensional harmonic oscillator problem using matrix methods.

Jan 1, 2010 Integrating the equation of motion for a one dimensional problem.
Solve for time for an arbitrary one dimensional potential.

Nov 26, 2009 Lagrangian and Euler-Lagrange equation evaluation for the spherical N-pendulum problem

Nov 4, 2009 Spherical polar pendulum for one and multiple masses (Take II)
The constraints required to derive the equations of motion from a bivector parameterized Lagrangian for the multiple spherical pendulum make the problem considerably more complicated than would be the case with a plain scalar parameterization. Take the previous multiple spherical pendulum and rework it with only scalar spherical polar angles. I later rework this once more removing all the geometric algebra, which simplifies it further.

Oct 27, 2009 Spherical polar pendulum for one and multiple masses, and multivector Euler-Lagrange formulation.
Derive the multivector Euler-Lagrange relationships. These were given in Doran/Lasenby but I did not understand it there. Apply these to the multiple spherical pendulum with the Lagrangian expressed in terms of a bivector angle containing all the phi dependence a scalar polar angle.

Sept 26, 2009 Hamiltonian notes.

Sept 22, 2009 Lorentz force from Lagrangian (non-covariant)
Show that the non-covariant Lagrangian from Jackson does produce the Lorentz force law (an exercise for the reader).

Sept 4, 2009 Translation and rotation Noether field currents.
Review Lagrangian field concepts. Derive the field versions of the Euler-Lagrange equations. Calculate the conserved current and conservation law, a divergence, for a Lagrangian with a single parameter symmetry (such as rotation or boost by a scalar angle or rapidity). Next, spacetime symmetries are considered, starting with the question of the symmetry existence, then a calculation of the canonical energy momentum tensor and its associated divergence relation. Next an attempt to use a similar procedure to calculate a conserved current for an incremental spacetime translation. A divergence relation is found, but it is not a conservation relationship having a nonzero difference of energy momentum tensors.

June 17, 2009 Comparison of two covariant Lorentz force Lagrangians
The Lorentz force Lagrangian for a single particle can be expressed in a quadratic fashion much like the classical Kinetic energy based Lagrangian. Compare to the proper time, non quadratic action.

June 5, 2009 Canonical energy momentum tensor and Lagrangian translation
Examine symmetries under translation and spacetime translation and relate to energy and momentum conservation where possible.

April 20, 2009 Tensor derivation of non-dual Maxwell equation from Lagrangian
A tensor only derivation.

April 15, 2009 Lorentz force Lagrangian with conjugate momentum
The Lagrangian can be expressed in a QM like form in terms of a sum of mechanical and electromagnetic momentum, mv + qA/c. The end result is the same and it works out to just be a factorization of the original Lorentz force covariant Lagrangian.

December 02, 2008 Compare some wave equation’s and their Lagrangians
A summary of some wave equation Lagrangians, including wave equations of quantum mechanics.

October 29, 2008 Field form of Noether’s Law

October 22, 2008 Lorentz transform Noether current for interaction Lagrangian

October 19, 2008 Lorentz Invariance of Maxwell Lagrangian

October 13, 2008 Euler Lagrange Equations

October 10, 2008 Derivation of Euler-Lagrange field equations
Derivation of the field form of the Euler Lagrange equations, with applications including Schrodinger’s and Klein-Gordan field equations

October 8, 2008 Revisit Lorentz force from Lagrangian

September 8, 2008 Direct variation of Maxwell equations

Sept 2, 2008 Attempts at solutions for some Goldstein Mechanics problems
Solutions to selected Goldstein Mechanics problems from chapter I and II.

Some of the Goldstein problems in chapter I were also in the Tong problem set. This is some remaining ones and a start at chapter II problems.

Problem 8 from Chapter I was never really completed in my first pass. It looks like I missed the Kinetic term in the Lagrangian too. The question of if angular momentum is conserved in that problem is considered in more detail, and a Noether’s derivation that is specific to the calculation of the conserved “current” for a rotational symmetry is performed. I’d be curious what attack on that question Goldstein was originally thinking of. Although I believe this Noether’s current treatment answers the question in full detail, since it wasn’t covered yet in the text, is there an easier way to get at the result?

September 1, 2008 Vector canonical momentum

August 30, 2008 Short metric tensor explanation
Metric tensor and Lorentz diagonality.

August 25, 2008 Solutions to David Tong’s mf1 Lagrangian problems

August 21, 2008 Covariant Lagrangian, and electrodynamic potential

August 9, 2008 Newton’s Law from Lagrangian

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new version of geometric algebra notes compilation

Posted by peeterjoot on October 25, 2011

Newly added to the compilation (version 10) since last update are:

April 30, 2011 A cylindrical Lienard-Wiechert potential calculation using multivector matrix products.

Dec 27, 2010 Vector form of Julia fractal.

Some typos are probably fixed too.

Changelog for version 9 was here.

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