Browsing by Author "Olver, Peter J."
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Item Algorithms for differential invariants of symmetry groups of differential equations(2005-10) Cheh, Jeongoo; Olver, Peter J.; Pohjanpelto, JuhaItem Canonical Forms and Conservation Laws in Linear Elastostatics(1997-10) Hatfield, Gary; Olver, Peter J.Item Canonical Forms and Integrability of BiHamiltonian Systems(1990-05) Olver, Peter J.Item Connections for general group actions(2003-09) Lewis, Debra; Nigam, Nilima; Olver, Peter J.Partial connections are (singular) differential systems generalizing classical connections on principal bundles, yielding analogous decompositions for manifolds with nonfree group actions. Connection forms are interpreted as maps determining projections of the tangent bundle onto the partial connection; this approach eliminates many of the complications arising from the presence of isotropy. A connection form taking values in the dual of the Lie algebra is smooth even at singular points of the action, while analogs of the classical algebra-valued connection form are necessarily discontinuous at such points. The curvature of a partial connection form can be defined under mild technical hypotheses; the interpretation of curvature as a measure of the lack of involutivity of the (partial) connection carries over to this general setting.Item Differential invariants of equi-affine surfaces(2007-02) Olver, Peter J.Item Differential invariants of maximally symmetric submanifolds(2008-09-16) Olver, Peter J.Let $G$ be a Lie group acting smoothly on a manifold $M$. A closed, nonsingular submanifold $S \subset M$ is called \is{maximally symmetric} if its symmetry subgroup $\sym S \subset G$ has the maximal possible dimension, namely $\dim \sym S = \dim S$, and hence $S = \sym S \cdot z_0$ is an orbit of $\sym S$. Maximally symmetric submanifolds are characterized by the property that all their differential invariants are constant. In this paper, we explain how to directly compute the numerical values of the differential invariants of a maximally symmetric submanifold from the infinitesimal generators of its symmetry group. The equivariant method of moving frames is applied to significantly simplify the resulting formulae. The method is illustrated by examples of curves and surfaces in various classical geometries.Item Dissipative decomposition of partial differential equations(1990-11) Olver, Peter J.; Shakiban, ChehrzadItem Equivalence of Higher Order Lagrangians I. Formulation and Reduction(1989) Kamran, Niky; Olver, Peter J.Item Equivalence of Higher Order Lagrangians II. The Cartan Form for Particle Lagrangians(1989) Hsu, Lucas; Kamran, Niky; Olver, Peter J.Item Existence and non-existence of solitary wave solutions to higher order model evolution equations(1991-05) Kichenassamy, Satyanad; Olver, Peter J.Item Generating differential invariants(2007-02) Olver, Peter J.Item Geodesic flow and two (super) component analog of the Camassa-Holm equation(2006-03) Guha, Partha; Olver, Peter J.Item Geometric foundations of numerical algorithms and symmetry(2000-04) Olver, Peter J.Item Geometric foundations of numerical algorithms and symmetry,(1999-02) Olver, Peter J.Item Geometric integration algorithms on homogeneous manifolds(2001-07) Lewis, Debra; Olver, Peter J.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.Item Ghost symmetries(2001-07) Olver, Peter J.; Sanders, Jan A.; Wang, Jing PingWe introduce the notion of a ghost symmetry for nonlocal differential equations. Ghosts are essential for maintaining the validity of the Jacobi identity for nonlocal vector fields.Item Internal, external and generalized symmetries(1990-11) Anderson, Ian M.; Kamran, Niky; Olver, Peter J.Item Invariant Euler-Lagrange equations and the invariant variational bicomplex(2000-12) Kogan, Irina A.; Olver, Peter J.In this paper, we derive an explicit group-invariant formula for the Euler-Lagrange equations associated with an invariant variational problem. The method relies on a group-invariant version of the variational bicomplex that is based on a general moving frame construction and is of independent interest.Item Invariant Theory, Equivalence Problems, and the Calculus of Variations(1989) Olver, Peter J.Item Lectures on moving frames(2009-01-21) Olver, Peter J.This article surveys the equivariant method of moving frames, along with a variety of applications to geometry, differential equations, computer vision, numerical analysis, the calculus of variations, and invariant curve flows.