Ian M. Anderson Copyright (c) 2013 All rights reserved. http://works.bepress.com/ian_anderson Recent documents in Ian M. Anderson en-us Sun, 24 Feb 2013 01:43:19 PST 3600 The Octonions and the Exceptional Lie Algebra G2 http://works.bepress.com/ian_anderson/76 http://works.bepress.com/ian_anderson/76 Fri, 22 Feb 2013 07:46:13 PST The octonions O are an 8-dimensional non-commutative, non-associative normed real algebra. The set of all derivations of O form a real Lie algebra. It is remarkable fact, first proved by E. Cartan in 1908, that the the derivation algebra of O is the compact form of the exceptional Lie algebra G2. In this worksheet we shall verify this result of Cartan and also show that the derivation algebra of the split octonions is the split real form of G2.

PDF and Maple worksheets can be downloaded from the links below.

]]> Ian M. Anderson New Symbolic Tools for Differential Geometry, Gravitation, and Field Theory (extended version) http://works.bepress.com/ian_anderson/75 http://works.bepress.com/ian_anderson/75 Fri, 22 Feb 2013 07:46:12 PST DifferentialGeometry is a Maple software package which symbolically performs fundamental operations of calculus on manifolds, differential geometry, tensor calculus, Lie algebras, Lie groups, transformation groups, jet spaces, and the variational calculus. These capabilities, combined with dramatic recent improvements in symbolic approaches to solving algebraic and differential equations, have allowed for development of powerful new tools for solving research problems in gravitation and field theory. The purpose of this paper is to describe some of these new tools and present some advanced applications involving: Killing vector fields and isometry groups, Killing tensors and other tensorial invariants, algebraic classification of curvature, and symmetry reduction of field equations.

]]> Charles G. Torre et al. New symbolic tools for differential geometry, gravitation, and field theory http://works.bepress.com/ian_anderson/74 http://works.bepress.com/ian_anderson/74 Fri, 22 Feb 2013 07:46:11 PST DifferentialGeometry is a Maple software package which symbolically performs fun- damental operations of calculus on manifolds, differential geometry, tensor calculus, spinor calculus, Lie algebras, Lie groups, transformation groups, jet spaces, and the variational calculus. These capabilities, combined with dramatic recent improve- ments in symbolic approaches to solving algebraic and differential equations, have allowed for development of powerful new tools for solving research problems in gravitation and field theory. The purpose of this paper is to describe some of these new tools and present some advanced applications involving: Killing vector fields and isometry groups, Killing tensors, algebraic classification of solutions of the Einstein equations, and symmetry reduction of field equations.

]]> Ian Anderson et al. Symmetric Criticality in Classical Field Theory http://works.bepress.com/ian_anderson/73 http://works.bepress.com/ian_anderson/73 Fri, 22 Feb 2013 07:46:09 PST This is a brief overview of work done by Ian Anderson, Mark Fels, and myself on symmetry reduction of Lagrangians and Euler-Lagrange equations, a subject closely related to Palais’ Principle of Symmetric Criticality. After providing a little history, I describe necessary and sufficient conditions on a group action such that reduction of a group-invariant Lagrangian by the symmetry group yields the correct symmetry-reduced Euler-Lagrange equations.

]]> Charles G. Torre et al. How to Create a Lie Algebra http://works.bepress.com/ian_anderson/72 http://works.bepress.com/ian_anderson/72 Fri, 22 Feb 2013 07:46:08 PST We show how to create a Lie algebra in Maple using three of the most common approaches: matrices, vector fields and structure equations. PDF and Maple worksheets can be downloaded from the links below.

]]> Ian M. Anderson How To Create The Quaternion & Octonion Algebras http://works.bepress.com/ian_anderson/71 http://works.bepress.com/ian_anderson/71 Fri, 22 Feb 2013 07:46:07 PST We show how to create the quaternion and octonion algebras with the DifferentialGeometry software. For each algebra, there is a split-form also available.

]]> Ian M. Anderson et al. How to Create a Clifford Algebra http://works.bepress.com/ian_anderson/70 http://works.bepress.com/ian_anderson/70 Fri, 22 Feb 2013 07:46:06 PST We show how to create a Clifford algebra in Maple using the DifferentialGeometry LieAlgebras package.

]]> Ian M. Anderson et al. Higher Order Symmetries of the KdV Equation http://works.bepress.com/ian_anderson/69 http://works.bepress.com/ian_anderson/69 Fri, 22 Feb 2013 07:46:05 PST In this worksheet we symbolically construct the formal inverse of the total derivative operator and use it to construct the recursion operator for the higher-order symmetries of the KdV equation. Using this recursion operator we generate the first 5 generalized symmetries of the KdV equation and verify that they all commute.

PDF and Maple worksheets can be downloaded from the links below.

]]> Ian M. Anderson Exterior Differential Systems with Symmetry http://works.bepress.com/ian_anderson/68 http://works.bepress.com/ian_anderson/68 Tue, 04 Oct 2011 13:32:36 PDT We use the theory of reduction of exterior differential systems with symmetry to study the problem of using a symmetry group of a differential equation to find noninvariant solutions.

]]> Ian M. Anderson et al. Perturbations of Conservation Laws in Field Theories http://works.bepress.com/ian_anderson/67 http://works.bepress.com/ian_anderson/67 Tue, 04 Oct 2011 13:32:33 PDT Conservation laws arising from Noether's theorem are varied to obtain a hierarchy of new conservation laws involving perturbations of the fields and their symmetries. The resulting conserved quantities, called Taub numbers, are obstructions to formal power series solutions of the field equations. The main results give conditions for the existence (conservation), for the triviality (vanishing), and for the gauge invariance of these Taub numbers. Gravitational and gauge fields are used to illustrate the theory; examples of other fields to which the theory can be extended are suggested.

]]> Ian M. Anderson et al. Tensorial Euler-Lagrange Expressions and Conservation Laws http://works.bepress.com/ian_anderson/66 http://works.bepress.com/ian_anderson/66 Tue, 04 Oct 2011 13:32:29 PDT Ian M. Anderson Group Invariant Solutions Without Transversality http://works.bepress.com/ian_anderson/65 http://works.bepress.com/ian_anderson/65 Tue, 04 Oct 2011 13:32:25 PDT We present a generalization of Lie's method for finding the group invariant solutions to a system of partial differential equations. Our generalization relaxes the standard transversality assumption and encompasses the common situation where the reduced differential equations for the group invariant solutions involve both fewer dependent and independent variables. The theoretical basis for our method is provided by a general existence theorem for the invariant sections, both local and global, of a bundle on which a finite dimensional Lie group acts. A simple and natural extension of our characterization of invariant sections leads to an intrinsic characterization of the reduced equations for the group invariant solutions for a system of differential equations. The characterization of both the invariant sections and the reduced equations are summarized schematically by the kinematic and dynamic reduction diagrams and are illustrated by a number of examples from fluid mechanics, harmonic maps, and general relativity. This work also provides the theoretical foundations for a further detailed study of the reduced equations for group invariant solutions.

]]> Ian M. Anderson et al. Asymptotic Conservation Laws in Classical Field Theory http://works.bepress.com/ian_anderson/64 http://works.bepress.com/ian_anderson/64 Tue, 04 Oct 2011 13:32:20 PDT A new, general, field theoretic approach to the derivation of asymptotic conservation laws is presented. In this approach asymptotic conservation laws are constructed directly from the field equations according to a universal prescription which does not rely upon the existence of Noether identities or any Lagrangian or Hamiltonian formalisms. The resulting general expressions of the conservation laws enjoy important invariance properties and synthesize all known asymptotic conservation laws, such as the Arnowitt-Deser-Misner energy in general relativity.

]]> Ian M. Anderson et al. Superposition Formulas for Darboux Integrable Exterior Differential Sys-tems http://works.bepress.com/ian_anderson/63 http://works.bepress.com/ian_anderson/63 Tue, 04 Oct 2011 13:32:16 PDT In this article we solve an inverse problem in the theory of quotients for differential equations. We characterize a family of exterior differential systems that can be written as a quotient of a direct sum of two associated systems that are constructed from the original. The fact that a system can be written as a quotient can be used to find the general solution to these equations. Some examples are given to demonstrate the theory.

]]> Ian M. Anderson et al. The Classification of Local Generalized Symmetries for the Einstein Equa- tions http://works.bepress.com/ian_anderson/62 http://works.bepress.com/ian_anderson/62 Tue, 04 Oct 2011 13:32:12 PDT Ian M. Anderson et al. The Principle of Minimal Gravitational Coupling http://works.bepress.com/ian_anderson/61 http://works.bepress.com/ian_anderson/61 Tue, 04 Oct 2011 13:32:09 PDT The principle of minimal gravitational coupling requires that the total Lagrangian for the field equations of general relativity consist of two additive parts, one part corresponding to the free gravitational Lagrangian and the other part to external source fields in curved spacetime. For source fields characterized by scalars, vectors and two-component spinors we prove that this additive decomposition of the total Lagrangian is an inevitable consequence of certain very general assumptions concerning the Euler-Lagrange expressions arising from the total Lagrangian.

]]> Ian M. Anderson Symmetry Reduction of Variational Bicomplexes and the Principle of Sym- metric Criticality http://works.bepress.com/ian_anderson/60 http://works.bepress.com/ian_anderson/60 Tue, 04 Oct 2011 13:32:06 PDT Consider a system of differential equations Δ = 0 which is invariant under a Lie group G of point transformations acting on the space E of independent and dependent variables. By a method due to Lie, the G invariant solutions of these differential equations are found by solving a reduced system of differential equations on the space Ē of invariants of G. In this paper we explore the relationship between the G invariant conservation laws and variational principles for the system of equations Δ = 0 and the conservation laws and variational principles for the reduced equations . This problem translates into one of constructing a certain cochain map ϱχ between the G invariant variational bicomplex for the infinite jet space on E and the free variational bicomplex for Ē. We prove that such a cochain map exists locally if and only if the relative Lie algebra cohomology condition is satisfied, where q is the orbit dimension of G, Γ the Lie algebra of vector fields on E which generate the infinitesimal action of G, and the linear isotropy subalgebra of Γ at . As a simple consequence we prove that the vanishing of is the only local obstruction to Palais' principle of symmetric criticality.

]]> Ian M. Anderson et al. A Characterization of the Einstein Tensor in Terms of Spinors http://works.bepress.com/ian_anderson/59 http://works.bepress.com/ian_anderson/59 Tue, 04 Oct 2011 13:32:02 PDT All tensors of contravariant rank two which are divergence‐free on one index, concomitants of a spinor field σiAX′ together with its first two partial derivatives, and scalars under spin transformations are constructed. The Einstein and metric tensors are the only candidates.

]]> Ian M. Anderson et al. Symmetries of the Einstein Equations http://works.bepress.com/ian_anderson/58 http://works.bepress.com/ian_anderson/58 Tue, 04 Oct 2011 13:31:58 PDT We classify all generalized symmetries of the vacuum Einstein equations in four spacetime dimensions. They consist of constant scalings of the metric, and of the infinitesimal action of generalized spacetime diffeomorphisms. Our results rule out a large class of possible ‘‘observables’’ for the gravitational field, and suggest that the vacuum Einstein equations are not integrable.

]]> Ian M. Anderson et al. On the Existence of Global Variational Principles http://works.bepress.com/ian_anderson/57 http://works.bepress.com/ian_anderson/57 Tue, 04 Oct 2011 13:31:55 PDT Ian M. Anderson et al.