Peeter Joot's (OLD) Blog.

Math, physics, perl, and programming obscurity.

An aborted attempt at a scientific style paper (geometric algebra N spherical pendulum)

Posted by peeterjoot on November 26, 2009

I’m not going to blog-ize this post since I’d have to first teach my latex to wordpress script to handle the theorem and subequations environment:

Matrix factorization of vectors in Geometric Algebra with application to the spherical N-pendulum problem

This was an attempt to cleanup my previous N spherical pendulum treatment (ie. equations of motion for an idealized chain). I’d used matrixes of multivector elements to arrive at an explicit expression for the kinetic energy for this system and then evaluated the Euler-Lagrange equations. I thought initially that this was pretty nifty, but as I was cleaning this up for an attempted arxiv post, I rudely realized that exactly the same result follows by factoring our a column vector of generalized coordinates (an angular velocity vector of sorts) from the scalar energy expressions (factoring a “one by one matrix” of scalar values into a product of matrixes).

No use of geometric algebra is required. Without the use of geometic algebra with its compact representation for the spherical parameterized points on the unit sphere, I don’t think I’d have even attempted this generalized problem. However, the idea that makes the problem tractable is completely independent of GA, and renders the paper as written just an obfuscated way to tackle it (since only a handful of people know this algebraic approach).

In case anybody is curious, here’s the abstract and introduction.

Abstract.

Geometric algebra provides a formalism that allows for coordinate and matrix free representations of vectors, linear transformations, and other mathematical structures. The replacement of matrix techniques with direct geometric algebra methods is a well studied field, but this need not be an exclusive replacement. While the geometric algebra multivector space does not constitute a field, operations on multivector element matrices can be defined in a consistent fashion. This use of geometric algebraic objects in matrices provides a hybrid mathematical framework with immediate applications to mathematical physics. Such an application is the dynamical examination of chain like structures, idealizable as an N mass pendulum with no planar constraints. The double planar pendulum and single mass spherical pendulum problems are well treated in Lagrangian physics texts, but due to complexity a similar treatment of the spherical N-pendulum problem is not pervasive. It will be shown that the multivector matrix technique lends itself nicely to this problem, allowing the complete specification of the equations of motion for this generalized pendulum system.

Introduction.

Derivation of the equations of motion for a planar motion constrained double pendulum system and a single spherical pendulum system are given as problems or examples in many texts covering Lagrangian mechanics. Setup of the Lagrangian, particularly an explicit specification of the system kinetic energy, is the difficult aspect of the multiple mass pendulum problem. Each mass in the system introduces additional interaction coupling terms, complicating the kinetic energy specification. In this paper, we utilize geometric (or Clifford) algebra to determine explicitly the Lagrangian for the spherical N pendulum system, and to evaluate the Euler-Lagrange equations for the system.

It is well known that the general specification of the kinetic energy for a system of independent point masses takes the form of a symmetric quadratic form \cite{goldstein1951cm} \cite{hestenes1999nfc}. However, actually calculating that energy explicitly for the general N-pendulum is likely thought too pedantic for even the most punishing instructor to inflict on students as a problem or example.

Considering this problem in a geometric algebra context motivates the introduction of multivector matrices. Given a one by one matrix containing a sum of vector terms, we can factor that matrix into a pair of $M \times 1$ and $1 \times M$ multivector matrices. In general there are many possible multivector factorizations of any given vector. For the pendulum problem all the required vector factorizations involve products with exponential rotation operators. This matrix factorization allows the velocity vector for each mass to be separated into a product containing one angular velocity coordinate vector and one multivector factor. Such a grouping can be used to tidily separate the kinetic energy into an explicit quadratic form, sandwiching a Hermitian multivector matrix between two vectors of generalized velocity coordinates.

While matrix algebra over the field of real numbers of complex numbers is familiar and well defined, geometric algebra multivector objects do not constitute a field, generally lacking both commutative multiplication and multiplicative inverses. This does not prevent a coherent definition of multivector matrix multiplication, provided that care is taken to retain appropriate order of element products. In the geometric algebra literature, there is significant focus on how to replace matrix and coordinate methods with geometric algebra, and less focus on how to use both together. A hybrid approach utilizing both matrix and geometric algebra has some elegance, and appears particularly well suited to the specific problem of the spherical N pendulum.

This paper is divided into two major sections. In the first, we start with a brief and minimal summary of required geometric algebra fundamentals and definitions. This includes the axioms, some nomenclature, and the use of one sided exponential rotation operators. To anybody already familiar with geometric algebra, this can be skipped or referred to only for notational conventions. Next, also in the first section, we define multivector matrix operations. A notion of Hermiticity useful for forming the quadratic forms used to express the kinetic energy for a multiple point mass system is defined here. A few examples of vector factorization into multivector matrix products are supplied for illustration purposes.

The second part of this paper applies the geometric algebra matrix techniques to the pendulum problem. For the single and double pendulum problems, already familiar to many, the methods and the results are compared to traditional coordinate geometry techniques used elsewhere. It is noteworthy that even the foundational geometric algebra mechanics text \cite{hestenes1999nfc} falls back to coordinate geometry for the planar double pendulum problem, instead of treating it in a way that is arguably more natural given the available tools.

Enroute to the Lagrangian we find an expression for the kinetic energy as a sum of bilinear forms, where the matrices used to express these forms are composed of pairs of block matrix products. This is surely a result that can be obtained by non geometric algebra methods, although how to obtain this result using alternate tools is not immediately obvious.

The end result of this paper is a complete and explicit specification of the Lagrangian and evaluation of the Euler-Lagrange equations for the chain-like N spherical pendulum system. While, this end result is free of geometric algebra, being nothing more than a non-linear set of coupled differential equations, it is believed that this geometric algebra matrix approach is an elegant and compact way to obtain it or to tackle similar problems. At the very least, this may prove to be one more useful addition to the physicist’s mathematical toolbox.

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