Browsing by Author "Lewis, Debra"

Now showing 1 - 7 of 7
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    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.
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    A geometric integration algorithm with applications to micromagnetics
    (2000-08) Lewis, Debra; Nigam, Nilima
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    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.
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    The heavy top: a geometric treatment
    (1991-10) Lewis, Debra; Ratiu, T.; Simo, J.C.; Marsden, J.E.
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    Lagrangian block diagonalization
    (1990-05) Lewis, Debra
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    Nonlinear stability of rotating pseudo-rigid bodies
    (1989) Lewis, Debra; Simo, J.C.
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    Recent progress in the use of geometric integration methods in micromagnetics and rigid body dynamics
    (2001-06) Lewis, Debra; Nigam, Nilima
    In this paper, we report further progress on our work on the use of Lie methods for integrating ordinary differential equations which evolve on manifolds. These algorithms better capture the qualitative behaviour of the trajectories since the numerical updates stay on the correct manifold. We study the effectiveness of higher order Lie methods in the context of rigid body dynamics, and for a problem in micromagnetics. This is work in progress.