sympy-0.6.4.tar.gz

************************************** Geometric Algebra Module for Sympy ************************************** :Author: Alan Bromborsky .. |release| replace:: 0.10 .. % Complete documentation on the extended LaTeX markup used for Python .. % documentation is available in ``Documenting Python'', which is part .. % of the standard documentation for Python. It may be found online .. % at: .. % .. % http://www.python.org/doc/current/doc/doc.html .. % \lstset{language=Python} .. % \input{macros} .. % This is a template for short or medium-size Python-related documents, .. % mostly notably the series of HOWTOs, but it can be used for any .. % document you like. .. % The title should be descriptive enough for people to be able to find .. % the relevant document. .. % Increment the release number whenever significant changes are made. .. % The author and/or editor can define 'significant' however they like. .. % At minimum, give your name and an email address. You can include a .. % snail-mail address if you like. .. % This makes the Abstract go on a separate page in the HTML version; .. % if a copyright notice is used, it should go immediately after this. .. % .. % \ifhtml .. % \chapter*{Front Matter\label{front}} .. % \fi .. % Copyright statement should go here, if needed. .. % ... .. % The abstract should be a paragraph or two long, and describe the .. % scope of the document. .. topic:: Abstract This document describes the implementation of a geometric algebra module in python that utilizes the :mod:`sympy` symbolic algebra library. The python module :mod:`GA` has been developed for coordinate free calculations using the operations (geometric, outer, and inner products etc.) of geometric algebra. The operations can be defined using a completely arbitrary pseudometric defined by the inner products of a set of arbitrary vectors or the pseudometric can be restricted to enforce orthogonality and signature constraints on the set of vectors. In addition the module includes the geometric, outer (curl) and inner (div) derivatives and the ability to define a curvilinear coordinate system. The module requires the numpy and the sympy modules. What is Geometric Algebra? ========================== Geometric algebra is the Clifford algebra of a real finite dimensional vector space or the algebra that results when a real finite dimensional vector space is extended with a product of vectors (geometric product) that is associative, left and right distributive, and yields a real number for the square (geometric product) of any vector [Hestenes,Lasenby]. The elements of the geometric algebra are called multivectors and consist of the linear combination of scalars, vectros, and the geometric product of two or more vectors. The additional axioms for the geometric algebra are that for any vectors :math:`a`, :math:`b`, and :math:`c` in the base vector space: .. math:: :nowrap: \begin{equation*} \begin{array}{c} a\lp bc \rp = \lp ab \rp c \\ a\lp b+c \rp = ab+ac \\ \lp a + b \rp c = ac+bc \\ aa = a^{2} \in \Re \end{array} \end{equation*} By induction these also apply to any multivectors. Several software packages for numerical geometric algebra calculations are available from Doran-Lazenby group and the Dorst group. Symbolic packages for Clifford algebra using orthongonal bases such as :math:`e_{i}e_{j}+e_{j}e_{i} = 2\eta_{ij}`, where :math:`\eta_{ij}` is a numeric array are available from the Doran-Lazenby group. The symbolic algebra module, :mod:`GA`, developed for python does not depend on an orthogonal basis representation, but rather is generated from a set of :math:`n` arbitrary symbolic vectors, :math:`a_{1},a_{2},\dots,a_{n}` and a symbolic pseudo-metric tensor :math:`g_{ij} = a_{i}\cdot a_{j}`. In order not to reinvent the wheel all scalar symbolic algebra is handled by the python module :mod:`sympy`. The basic geometic algebra operations will be implemented in python by defining a multivector class, MV, and overloading the python operators in Table :ref:`1 <table1>` where ``A`` and ``B`` are any two multivectors. .. _table1: +-----------+-----------------------------------------------+ | Operation | Result | +===========+===============================================+ | ``A+B`` | sum of multivectors or multivector and scalar | +-----------+-----------------------------------------------+ | ``A-B`` | difference of multivectors or multivector | | | and scalar | +-----------+-----------------------------------------------+ | ``A*B`` | geometric product or multiplication by scalar | +-----------+-----------------------------------------------+ | ``A^B`` | outer product of multivectors | +-----------+-----------------------------------------------+ | ``A|B`` | inner product of multivectors | +-----------+-----------------------------------------------+ Table :ref:`1 <table1>`. Multivector operations for symbolicGA Note that the operator order precedence is determined by python and is not neccessarily that used by geometric algebra. Always use parenthesis in python expressions containing ``:math:`\W``` and/or ``|``. .. _vbm: Vector Basis and Metric ======================= The two structures that define the :class:`MV` (multivector) class are the symbolic basis vectors and the symbolic pseudometric. The symbolic basis vectors are input as a string with the symbol name separated by spaces. For example if we are calculating the geometric algebra of a system with three vectors that we wish to denote as ``a0``, ``a1``, and ``a2`` we would define the string variable: ``basis = 'a0 a1 a2'`` that would be input into the multivector setup function. The next step would be to define the symbolic pseudometric for the geometric algebra of the basis we have defined. The default pseudometric is the most general and is the matrix of the following symbols .. _eq1: .. math:: :label: 1 :nowrap: \begin{equation*} g = \lbrk \begin{array}{ccc} a0**2 & (a0.a1) & (a0.a2) \\ (a0.a1) & a1**2 & (a1.a2) \\ (a0.a2) & (a1.a2) & a2**2 \\ \end{array} \rbrk \end{equation*} where each of the :math:`g_{ij}` is a symbol representing all of the dot products of the basis vectors. Note that the symbols are named so that :math:`g_{ij} = g_{ji}` since for the symbol function :math:`(a0.a1) \ne (a1.a0)`. Note that the strings shown in equation :ref:`1 <eq1>` are only used when the values of :math:`g_{ij}` are output (printed). In the :mod:`GA` module (library) the :math:`g_{ij}` symbols are stored in a static member list of the multivector class :class:`MV` as the double list *MV.metric* (:math:`g_{ij} = ` *MV.metric[i][j]*). The default definition of :math:`g` can be overwritten by specifying a string that will define :math:`g`. As an example consider a symbolic representation for conformal geometry. Define for a basis ``basis = 'a0 a1 a2 n nbar'`` and for a metric ``metric = '# # # 0 0, # # # 0 0, # # # 0 0, 0 0 0 0 2, 0 0 0 2 0'`` then calling :func:`MV.setup(basis,metric)` would initialize .. math:: :nowrap: \begin{equation*} g = \lbrk \begin{array}{ccccc} a0**2 & (a0.a1) & (a0.a2) & 0 & 0\\ (a0.a1) & a1**2 & (a1.a2) & 0 & 0\\ (a0.a2) & (a1.a2) & a2**2 & 0 & 0 \\ 0 & 0 & 0 & 0 & 2 \\ 0 & 0 & 0 & 2 & 0 \end{array} \rbrk \end{equation*} Here we have specified that :math:`n` and :math:`nbar` are orthonal to all the :math:`a`'s, :math:`n**2 = nbar**2 = 0`, and :math:`(n.nbar) = 2`. Using :math:`\#` in the metric definition string just tells the program to use the default symbol for that value. When ``MV.setup`` is called multivector representations of the basis local to the program are instantiated. For our first example that means that the symbolic vectors named ``a0``, ``a1``, and ``a2`` are c