A multivector data structure for differential forms and equations

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Abstract

We use tools from algebraic topology to show that a class of structural differential equations may be represented combinatorially, and thus, by a computer data structure. In particular, every differential k-form may be represented by a formal k-cochain over a cellular structure that we call a starplex, and exterior differentiation is equivalent to the coboundary operation on the corresponding k-cochain. Furthermore, there is a one-to-one correspondence between this model and the classical finite cellular model supported by the Generalized Stokes’ Theorem and translation between the two models can be completely automated.

Our results point the way to a common combinatorial and data structure well-suited for a physical modeling computer algebra that unifies finite and infinitesimal, symbolic and numeric, geometric and physical descriptions of distributed phenomena. We illustrate the advantages of our approach by a prototype interactive physics editor that uses the computer algebra to automatically translate intuitive geometrical/physical descriptions of balance conditions created by the user into the corresponding symbolic differential and integral equations.

Introduction

Much of scientific computing and engineering analysis deals with physical phenomena that are distributed in space and time and are typically modeled by partial differential equations. Symbolic descriptions of partial differential equations have evolved over the years from the classical notations used by Leibnitz and Newton to vector and tensor calculus, and later to a more geometric calculus of differential forms and geometric algebra of multivectors. Each transformation in the evolution of the languages for differential equations is marked by the desire to make the language simpler, more precise, and/or more universal. Geometry and topology play a special role in this quest for the ideal language for several reasons:

  • All symbolic quantities, but particularly spatial and temporal variables in differential equations, have a well-defined type.

  • Most differential equations are derived by postulating physical laws for small regions of assumed dimension and shape.

  • Many differential equations are instances of the same fundamental structural laws or theorems, differing from each other only by the choice of coordinates, dimensionality, or selected labels.

Without going into widely discussed advantages and disadvantages of various symbolic descriptions, we note that the above geometric and topological information — quantity type, dimension, shape, and the fundamental law — is implied by the usual symbolic forms and notations, but it is not represented explicitly by any of them (with the notable exception of dimension in differential forms). Thus, the types of variables are usually implied by the associated units and operations; symbols x,y,r,θ, etc. signal the choice of particular coordinate systems; and significant specialized experience is often needed in order to interpret and to relate given differential equations geometrically. The lack of an explicit structure to maintain the geometric and topological information manifests itself in several important ways.

Proliferation of models. Fig. 1 illustrates a typical sequence of steps in a traditional modeling process, starting with a model formulation and ending with a numerical approximation based on a spatial discretization. Such physical models of various distributed phenomena are being derived on a case-by-case basis, usually manually, and often repeating or rearranging a sequence of steps differing only in variable naming, choice of coordinate systems, or dimension. The underlying assumptions and models may be difficult to reconcile and relate, leading to artificial distinctions and inter-disciplinary barriers.

Communication barriers. The symbolic notation of the differential and integral equations is reminiscent of Latin in the Middle Ages: on one hand, the commonly accepted and the standardized language facilitates broad scientific exchanges and enables great progress; on the other hand, its apparent complexity and narrow specialization (usually implied by the notation) effectively limits many scientists and engineers from participating in rigorous modeling and analysis of spatially distributed phenomena. Most physical laws follow from experimental observations and can be expressed algebraically or geometrically (right middle of Fig. 1), but they must be translated into the languages of differential equations (left middle of Fig. 1) before they may be abstracted, modeled on a computer, and eventually solved.

Segregation of models and solutions. Few differential equations can be solved exactly, and most common solution procedures rely on spatial and temporal discretization techniques that associate approximate physical quantities with the resulting geometric data structures (bottom row of Fig. 1). Such data structures are usually associated with the solution method and not the first-principles differential model, and often appear to be incompatible with the form and structure of the original differential equations. Once again, this forces a case-by-case analysis, and prevents automatic translation of the analytical models into numerical approximation schemes.

In this paper, we use tools from algebraic topology to show that a class of structural differential equations may be represented combinatorially, and thus, by a computer data structure. Furthermore, there is a one-to-one correspondence between the new model and the classical finite cellular model supported by the Generalized Stokes’ Theorem, and the translation between the two models can be completely automated. Our results point the way to a common combinatorial and data structure well-suited for a physical modeling computer algebra that unifies finite and infinitesimal, symbolic and numeric, geometric and physical descriptions of distributed phenomena. We illustrate the advantages of our approach by a prototype interactive physics editor program that accepts from the user intuitive geometrical/physical descriptions for the balance conditions, which do not require knowledge of associated symbolic descriptions, and automatically translates them into the equivalent symbolic differential and integral equations.

The quest for classification and unification of physical quantities and theories has always been a central theme in science. James Clerk Maxwell was one of the first to express the desire to establish a formal analogy between various physical quantities based on their mathematical form [38]. The study of physical analogies has continued and intensified in the 20th century (for example see [56]) fueled by developments in engineering and mathematics that we briefly summarize below.

Advances in electrical systems analysis and manufacturing led many to explore the possibility of extending this success to other physical, and in particular, mechanical domains [41], [43]. Nickle was one of the first to map mechanical dynamical systems into ‘equivalent’ analog electrical circuits whose response predicted that of the modeled mechanical system [41]. Similar ideas led Henry Paynter to developing his language of bond-graphs that describe multiple domain lumped-parameter systems and transformations between them in a graph language that can be automatically translated into a system of ordinary differential equations [46]. This electrical-network approach to modeling and simulation of mechanical systems culminated in the work of Gabriel Kron, a controversial electrical engineer, who showed how distributed mechanical systems may be modeled by multi-dimensional electrical networks and proposed an efficient method of ‘diakoptics’ for solving them [33], [34]. Kron’s disciples included Branin who recognized that the versatility of electrical networks is an indication of the combinatorial nature of the classical vector calculus [5], and Roth who identified the algebraic topology as that common structure responsible for the apparent analogies in physical theories [48], [49].

The quest for unification of physical theories has proceeded in parallel with the development of mathematical theories aimed at identifying a common geometric structure and symbolic language for calculus. In the mid 19th century, Hermann Grassmann and Sir William Hamilton came up with two different versions of a geometric calculus: Grassmann’s so-called “calculus of extensions” and Hamilton’s algebra of quaternions. Although Grassmann attempted to fit quaternions into the calculus of extensions on his own, William Clifford is generally credited with unifying the two, and geometric algebra is commonly called Clifford Algebra in recognition of the achievement [27], [42]. The significance of Clifford’s Algebra was not immediately recognized and advocates of Grassmann and Hamilton followed along separate paths. Hamilton’s work was eventually simplified by Josiah Gibbs and Oliver Heaviside into the modern version of vector calculus [47], although a conflicting account claims that the simplification is from Grassmann’s work [4]. What does appear certain is that Grassmann influenced Elie Cartan in his development of exterior differential forms [42], [51] that are becoming an increasingly accepted language for mathematics and physics at all levels [1].

One important advantage of differential forms over the usual expressions in vector calculus is that every form comes with an explicit declaration of dimensionality. This eliminates the possibility of symbolic manipulations which are possible in the vector calculus, but have no physical meaning (such as curl(curlf)). A related advantage of differential forms is that, in contrast to vector calculus, differential forms generalize for any dimension n [58].

A movement towards further generalization appears to be taking shape. The geometric algebra [22], [23] is being advocated as a ‘universal’ language for dealing with a variety of physical applications. Example problems in classical mechanics [24], electromagnetism, quantum mechanics and general relativity [2] have been published in the past decade or so to indicate the broad applicability of this mathematical tool.

In his landmark manuscript [54], Tonti bridged the artificial gap separating engineering and mathematics by showing that most physical models may be classified based on their algebraic topological structure, with cochains (see Appendix A) as the discrete analogues of differential forms ([54], Chapter 7). Palmer and Shapiro [45] built on Tonti’s work to propose a universal combinatorial structure and language for modeling and analysis of physical systems.

However, none of the proposed combinatorial structures are suitable for explicitly maintaining the necessary geometric and topological information in differential (or infinitesimal) statements of physical behavior. The main contribution of this work is to close this void by defining a structure which does maintain this information.

Equations of differential forms contain the same information as more traditional (vector calculus) differential equations [12], along with the explicit dimensionality of elements. In this paper, we show that differential forms (and therefore other traditional symbolic differential expressions) may be represented using an explicit cellular structure that can be informally described as a star pseudo-complex. It is a proper subset of the usual cubical cell complex that is commonly used in spatial discretizations and approximations of distributed phenomena. Differential forms may be represented by cochains on such a structure, (exterior) differentiation corresponds to a modified coboundary operation, and the translation between infinitesimal differential and finite integral statements may be completely automated via the classical Stokes’ Theorem and its dual described in Section 5.2.

The combinatorial structure of the constructed model explicates the known analogies between distinct physical domains and establishes a direct hierarchy between various models in terms of the assumed dimension and variable types. The described model immediately translates into a straightforward and intuitive computer data structure that enforces the strong typing of variables in the model and can be manipulated by users with minimal understanding of the underlying mathematics.

We eschew the use of geometric algebra in this paper, but we explain in Section 3.3 that our data structure is well-suited for representing the fundamental objects of geometric algebra, called multivectors. Accordingly, we use the term ‘multivector data structure,’ even though in this paper, we restrict its use to differential forms and equations. That said, we would like the reader to recognize that multivectors are suitable for representing most other languages of mathematical physics. As described in [13], [14], [23], [24], [25] among others, these geometric objects facilitate the expression of laws currently written with tensors, differential forms, spinors, twistors and others. So our data structure is just as well-positioned to express tensorial descriptions as it is in our current presentation of differential forms; such versatility suggests that the multivector data structure is a powerful and promising choice for computer algebra systems applied to problems of mathematical physics.

The rest of the paper is organized as follows. The next section, Section 2, provides a cursory overview of k-vectors and their dual quantities, differential forms. Section 3 proposes an alternative representation for multivectors and differential forms which naturally leads to an explicit data structure. We then show how some of the typical differential form operations may be algorithmically performed on our proposed data structure in Section 4. Section 5 shows how translation between finite and infinitesimal domains — steps necessary according to the middle and bottom rows of Fig. 1 — may also be performed algorithmically on the multivector data structure. The concluding section, Section 6, explains the significance and advantages of modeling with the multivector structure, and discusses a prototype implementation of an interactive physics editor.

Section snippets

Mathematical preliminaries

The process in Fig. 1 may be described in several formally equivalent ways. For the purposes of this paper, it is convenient to recognize that infinitesimal control elements of dimension k are k-vectors. For example, a line segment is a 1-vector, a three-dimensional (3D) cube is a 3-vector, and so on. The physical quantities that are attached to the control elements are usually described by functions of k-vectors, called differential forms, representing distributions of the assumed quantities

The multivector structure

The precise relationship between k-forms, k-vectors and Fig. 1 may not be obvious for several reasons. Both the control element in Fig. 1 and its representation by a k-vector appear distinctly homogeneous in dimension k. The usual coordinate-free views of k-vectors reflected in images of Fig. 2 and of the dual k-forms in Fig. 4 de-emphasize the combinatorial nature of the respective linear spaces, obscuring the relationship between vectors and forms of different dimensions. This is in sharp

Operations and physical laws

We asserted earlier that the multivector structure will be useful in unifying and simplifying manipulation of physical models; now we supplement the multivector structure with several fundamental operations and illustrate how they can be used for formulating and expressing physical laws. We focus on arguably one of the most fundamental physical laws — that of balance of an assumed physical quantity.

Unification

By now, it should be apparent that every physical balance law may be described in two distinct ways: in a finite integral sense on a finite cell complex, or in an infinitesimal differential sense on the starplex. Returning to Fig. 1, we see that the transition between the finite and infinitesimal models occurs twice in a typical modeling scenario. The first time, in the middle row, a finite unit size control element is shrunk down through some limiting process into an infinitesimal model to

Advantages of the multivector structure

We now revisit the modeling process illustrated in Fig. 1, to see how it may be supported by the multivector data structure. Because the whole process may now be restated in terms of the same combinatorial structure, the modeling sequence may be ‘collapsed’ and represented by a set of transformations operating on the same structure, as shown in Fig. 13. At the heart of the structure is a finite n-dimensional cubical cell complex and the corresponding starplex. Coordinate systems are effected by

Acknowledgements

This research was supported in part by the National Science Foundation, NSF grants DMI-9502728 and DMI-9522806. J. ∼Chard was also supported by the Department of Education fellowship under P200A70715 and NSF fellowship as part of DMS-9256439. The authors are grateful to Professor Enzo Tonti of the University of Trieste for critical reading of an earlier version of the manuscript and suggesting a number of corrections. Responsibility for any errors or omissions lies solely with the authors.

References (60)

  • G.E. Bredon, Topology and Geometry, Springer, New York,...
  • E. Brisson, Representation of d-dimensional geometric objects, Ph.D. Thesis, University of Washington, Seattle, WA,...
  • W.L. Burke, Applied Differential Geometry, Cambridge University Press, Cambridge, UK,...
  • R. Courant, D. Hilbert, Methods of Mathematical Physics, Wiley, New York,...
  • R. Courant, F. John, Introduction to Calculus and Analysis, Wiley, New York,...
  • J.F. Cremer et al.

    Creating scientific software

    Trans. Soc. Comput. Simul. Int.

    (1997)
  • R.W.R. Darling, Differential Forms and Connections, Cambridge University Press, Cambridge, UK,...
  • C. Doran et al.

    Lie groups as spin groups

    J. Math. Phys.

    (1993)
  • C.J.L. Doran, Geometric algebra and its application to mathematical physics, Ph.D. Thesis, University of Cambridge,...
  • H.M. Edwards, Advanced Calculus: A Differential Forms Approach, 3rd Edition, Prentice-Hall, Englewood Cliffs, NJ,...
  • H. Flanders, Differential Forms with Applications to the Physical Sciences, Dover, New York,...
  • R.W. Fox, A.T. McDonald, Introduction to Fluid Mechanics, Wiley, New York,...
  • V. Guillemin, A. Pollack, Differential Topology, Prentice-Hall, Englewood Cliffs, NJ,...
  • S. Gull, A. Lasenby, C. Doran, Imaginary numbers are not real — the geometric algebra of spacetime, Technical report,...
  • S. Gull, A. Lasenby, C. Doran, Geometric algebra, spacetime physics and gravitation, in: O. Lahav, E. Terlevick, R.J....
  • D. Hestenes, Spacetime Algebra, Gordon and Breach, London,...
  • D. Hestenes, Clifford Algebra to Geometric Calculus, A Unified Language for Mathematics and Physics, Reidel, Dordrecht,...
  • D. Hestenes, New Foundations for Classical Mechanics, Reidel, Dordrecht,...
  • D. Hestenes, A unified language for mathematics and physics, in: J.S.R. Chisholm, A.K. Common (Eds.), Clifford Algebras...
  • D. Hestenes, Differential forms in geometric calculus, in: F. Brackx, et al. (Eds.), Clifford Algebras and their...
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