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34
Discrete mechanics and variational integrators
 Acta Numer
, 2001
"... This paper gives a review of integration algorithms for finite dimensional mechanical systems that are based on discrete variational principles. The variational technique gives a unified treatment of many symplectic schemes, including those of higher order, as well as a natural treatment of the disc ..."
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Cited by 160 (30 self)
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This paper gives a review of integration algorithms for finite dimensional mechanical systems that are based on discrete variational principles. The variational technique gives a unified treatment of many symplectic schemes, including those of higher order, as well as a natural treatment of the discrete Noether theorem. The approach also allows us to include forces, dissipation and constraints in a natural way. Amongst the many specific schemes treated as examples, the Verlet, SHAKE, RATTLE, Newmark, and the symplectic
Joint invariant signatures
 Found. Comput. Math
, 1999
"... Dedicated to the memory of Gian–Carlo Rota Abstract. A new, algorithmic theory of moving frames is applied to classify joint invariants and joint differential invariants of transformation groups. Equivalence and symmetry properties of submanifolds are completely determined by their joint signatures, ..."
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Cited by 32 (20 self)
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Dedicated to the memory of Gian–Carlo Rota Abstract. A new, algorithmic theory of moving frames is applied to classify joint invariants and joint differential invariants of transformation groups. Equivalence and symmetry properties of submanifolds are completely determined by their joint signatures, which are parametrized by a suitable collection of joint invariants and/or joint differential invariants. A variety of fundamental geometric examples are developed in detail. Applications to object recognition problems in computer vision and the design of invariant numerical approximations are indicated.
Geometric foundations of numerical algorithms and symmetry
 Appl. Alg. Engin. Commun. Comput
"... Abstract. This paper outlines a new general construction, named “multispace”, that forms the proper geometrical foundation for the numerical analysis of differential equations — in direct analogy with the role played by jet space as the basic object underlying the geometry of differential equations ..."
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Cited by 24 (15 self)
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Abstract. This paper outlines a new general construction, named “multispace”, that forms the proper geometrical foundation for the numerical analysis of differential equations — in direct analogy with the role played by jet space as the basic object underlying the geometry of differential equations. Application of the theory of moving frames leads to a general framework for constructing symmetrypreserving numerical approximations to differential invariants and invariant differential equations.
Geometric Integrators for ODEs
 J. Phys. A
, 2006
"... Abstract. Geometric integration is the numerical integration of a differential equation, while preserving one or more of its “geometric ” properties exactly, i.e. to within roundoff error. Many of these geometric properties are of crucial importance in physical applications: preservation of energy, ..."
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Cited by 18 (5 self)
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Abstract. Geometric integration is the numerical integration of a differential equation, while preserving one or more of its “geometric ” properties exactly, i.e. to within roundoff error. Many of these geometric properties are of crucial importance in physical applications: preservation of energy, momentum, angular momentum, phase space volume, symmetries, timereversal symmetry, symplectic structure and dissipation are examples. In this paper we present a survey of geometric numerical integration methods for ordinary differential equations. Our aim has been to make the review of use for both the novice and the more experienced practitioner interested in the new developments and directions of the past decade. To this end, the reader who is interested in reading up on detailed technicalities will be provided with numerous signposts to the relevant literature. Geometric Integrators for ODEs 2 1.
Moving frames and singularities of prolonged group actions
 Selecta Math. (N.S
"... Abstract. The prolongation of a transformation group to jet bundles forms the geometric foundation underlying Lie’s theory of symmetry groups of differential equations, the theory of differential invariants, and the Cartan theory of moving frames. Recent developments in the moving frame theory have ..."
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Cited by 17 (12 self)
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Abstract. The prolongation of a transformation group to jet bundles forms the geometric foundation underlying Lie’s theory of symmetry groups of differential equations, the theory of differential invariants, and the Cartan theory of moving frames. Recent developments in the moving frame theory have necessitated a detailed understanding of the geometry of prolonged transformation groups. This paper begins with a basic review of moving frames, and then focuses on the study of both regular and singular prolonged group orbits. Highlights include a corrected version of the basic stabilization theorem, a discussion of “totally singular points, ” and geometric and algebraic characterizations of totally singular submanifolds, which are those that admit no moving frame. In addition to applications to the method of moving frames, the paper includes a generalized Wronskian lemma for vectorvalued functions, and methods for the solution to Lie determinant equations.
On Cayleytransform methods for the discretization of Liegroup equations
 FOUND. COMPUT. MATH
, 1999
"... In this paper we develop in a systematic manner the theory of timestepping methods based on the Cayley transform. Such methods can be applied to discretise differential equations that evolve in some Lie groups, in particular in the orthogonal group and the symplectic group. Unlike many other Liegr ..."
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Cited by 16 (4 self)
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In this paper we develop in a systematic manner the theory of timestepping methods based on the Cayley transform. Such methods can be applied to discretise differential equations that evolve in some Lie groups, in particular in the orthogonal group and the symplectic group. Unlike many other Liegroup solvers, they do not require the evaluation of matrix exponentials. Similarly to the theory of Magnus expansions in (Iserles & Nørsett 1999), we identify terms in a Cayley expansion with rooted trees, which can be constructed recursively. Each such term is an integral over a polytope but all such integrals can be evaluated to high order by using special quadrature formulae similar to the construction in (Iserles & Nrsett 1999). Truncated Cayley expansions (with exact integrals) need not be timesymmetric, hence the method does not display the usual advantages associated with time symmetry, e.g. even order of approximation. However, time symmetry (with its attendant benefits) is attained when exact integrals are replaced by certain quadrature formulae. 1 Quadratic Lie groups The theme of this paper is geometric integration: numerical discretization of differential equations that respects their underlying geometry. It is increasingly recognised by numerical analysts and users of computational methods alike that geometric integration often represents a highly efficient and precise means toward obtaining a numerical solution, whilst retaining important qualitative attributes of the differential system (Budd & Iserles 1999). Large number of differential equations with a wide range of practical applications evolve on Lie groups G = fA 2 GLn (R) : AJA where GLn (R) is the group of all n \Theta n nonsingular real matrices and J 2 GLn (R) is given. (We refer the reader to (Cart...
A survey of moving frames
 Computer Algebra and Geometric Algebra with Applications. Volume 3519 of Lecture Notes in Computer Science, 105–138
, 2005
"... Abstract. This article presents the equivariant method of moving frames for finitedimensional Lie group actions, surveying a variety of applications, including geometry, differential equations, computer vision, numerical analysis, the calculus of variations, and invariant flows. 1. Introduction. Acc ..."
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Cited by 15 (3 self)
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Abstract. This article presents the equivariant method of moving frames for finitedimensional Lie group actions, surveying a variety of applications, including geometry, differential equations, computer vision, numerical analysis, the calculus of variations, and invariant flows. 1. Introduction. According to Akivis, [1], the method of moving frames originates in work of the Estonian mathematician Martin Bartels (1769–1836), a teacher of both Gauss and Lobachevsky. The field is most closely associated with Élie Cartan, [21], who forged earlier contributions by Darboux, Frenet, Serret, and Cotton into a powerful tool for analyzing the geometric
Geometric integration algorithms on homogeneous manifolds
 Foundations of Computational Mathematics
, 2002
"... Given an ordinary differential equation on a homogeneous manifold, one can construct a “geometric integrator ” by determining a compatible ordinary differential equation on the associated Lie group, using a Lie group integration scheme to construct a discrete time approximation of the solution curve ..."
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Cited by 10 (3 self)
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Given an ordinary differential equation on a homogeneous manifold, one can construct a “geometric integrator ” by determining a compatible ordinary differential equation on the associated Lie group, using a Lie group integration scheme to construct a discrete time approximation of the solution curves in the group, and then mapping the discrete trajectories onto the homogeneous manifold using the group action. If the points of the manifold have continuous isotropy, a vector field on the manifold determines a continuous family of vector fields on the group, typically with distinct discretizations. If sufficient isotropy is present, an appropriate choice of vector field can yield improved capture of key features of the original system. In particular, if the algebra of the group is “full”, then the order of accuracy of orbit capture (i.e. approximation of trajectories modulo time reparametrization) within a specified family of integration schemes can be increased by an appropriate choice of isotropy element. We illustrate the approach developed here with comparisons of several integration schemes for the reduced rigid body equations on the sphere. 1 Introduction. Geometric integration techniques have become increasingly popular in the modern approach to numerical
What kinds of dynamics are there? Lie pseudogroups, dynamical systems, and geometric integration
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The Magnus expansion and some of its applications
, 2008
"... Approximate resolution of linear systems of differential equations with varying coefficients is a recurrent problem shared by a number of scientific and engineering areas, ranging from Quantum Mechanics to Control Theory. When formulated in operator or matrix form, the Magnus expansion furnishes an ..."
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Cited by 8 (0 self)
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Approximate resolution of linear systems of differential equations with varying coefficients is a recurrent problem shared by a number of scientific and engineering areas, ranging from Quantum Mechanics to Control Theory. When formulated in operator or matrix form, the Magnus expansion furnishes an elegant setting to built up approximate exponential representations of the solution of the system. It provides a power series expansion for the corresponding exponent and is sometimes referred to as TimeDependent Exponential Perturbation Theory. Every Magnus approximant corresponds in Perturbation Theory to a partial resummation of infinite terms with the important additional property of preserving at any order certain symmetries of the exact solution. The goal of this review is threefold. First, to collect a number of developments scattered through half a century of scientific literature on Magnus expansion. They concern the methods for the generation of terms in the expansion, estimates of the radius of convergence of the series, generalizations and related nonperturbative