13,639 research outputs found
Principal Geodesic Dynamics
International audienceThis paper presents a new integration of a data-driven approach using dimension reduction and a physically-based simulation for real-time character animation. We exploit Lie group statistical analysis techniques (Principal Geodesic Analysis, PGA) to approximate the pose manifold of a motion capture sequence by a reduced set of pose geodesics. We integrate this kinematic parametrization into a physically-based animation approach of virtual characters, by using the PGA-reduced parametrization directly as generalized coordinates of a Lagrangian formulation of mechanics. In order to achieve real-time without sacrificing stability, we derive an explicit time integrator by approximating existing variational integrators. Finally, we test our approach in task-space motion control. By formulating both physical simulation and inverse kinematics time stepping schemes as two quadratic programs, we propose a features-based control algorithm that interpolates between the two metrics. This allows for an intuitive trade-off between realistic physical simulation and controllable kinematic manipulation
Conformal scattering on the Schwarzschild metric
We show that existing decay results for scalar fields on the Schwarzschild
metric are sufficient to obtain a conformal scattering theory. Then we
re-interpret this as an analytic scattering theory defined in terms of wave
operators, with an explicit comparison dynamics associated with the principal
null geodesic congruences. The case of the Kerr metric is also discussed.Comment: 36 pages, 6 figures. From the first version, recent references have
been added and the discussion has been modified to take the new references
into account. To appear in Annales de l'Institut Fourie
Long-time principal geodesic analysis in director-based dynamics of hybrid mechanical systems
In this article, we investigate an extended version of principal geodesic analysis for the unit sphere S2 and the special orthogonal group SO(3). In contrast to prior work, we address the construction of long-time smooth lifts of possibly non-localized data across branches of the respective logarithm maps. To this end, we pay special attention to certain critical numerical aspects such as singularities and their consequences on the numerical accuracy. Moreover, we apply principal geodesic analysis to investigate the behavior of several mechanical systems that are very rich in dynamics. The examples chosen are computationally modeled by employing a director-based formulation for rigid and flexible mechanical systems. Such a formulation allows to investigate our algorithms in a direct manner while avoiding the introduction of additional sources of error that are unrelated to principal geodesic analysis. Finally, we test our numerical machinery with the examples and, at the same time, we gain deeper insight into their dynamical behavior
Geometric dynamics on the automorphism group of principal bundles: geodesic flows, dual pairs and chromomorphism groups
We formulate Euler-Poincar\'e equations on the Lie group Aut(P) of
automorphisms of a principal bundle P. The corresponding flows are referred to
as EPAut flows. We mainly focus on geodesic flows associated to Lagrangians of
Kaluza-Klein type. In the special case of a trivial bundle P, we identify
geodesics on certain infinite-dimensional semidirect-product Lie groups that
emerge naturally from the construction. This approach leads naturally to a dual
pair structure containing \delta-like momentum map solutions that extend
previous results on geodesic flows on the diffeomorphism group (EPDiff). In the
second part, we consider incompressible flows on the Lie group of
volume-preserving automorphisms of a principal bundle. In this context, the
dual pair construction requires the definition of chromomorphism groups, i.e.
suitable Lie group extensions generalizing the quantomorphism group.Comment: 52 pages; revised versio
Un-reduction
This paper provides a full geometric development of a new technique called
un-reduction, for dealing with dynamics and optimal control problems posed on
spaces that are unwieldy for numerical implementation. The technique, which was
originally concieved for an application to image dynamics, uses Lagrangian
reduction by symmetry in reverse. A deeper understanding of un-reduction leads
to new developments in image matching which serve to illustrate the
mathematical power of the technique.Comment: 25 pages, revised versio
Generalized Calogero-Moser-Sutherland models from geodesic motion on GL(n, R) group manifold
It is shown that geodesic motion on the GL(n, R) group manifold endowed with
the bi-invariant metric d s^2 = tr(g^{-1} d g)^2 corresponds to a
generalization of the hyperbolic n-particle Calogero-Moser-Sutherland model. In
particular, considering the motion on Principal orbit stratum of the SO(n, R)
group action, we arrive at dynamics of a generalized n-particle
Calogero-Moser-Sutherland system with two types of internal degrees of freedom
obeying SO(n, R) \bigoplus SO(n, R) algebra. For the Singular orbit strata of
SO(n, R) group action the geodesic motion corresponds to certain deformations
of the Calogero-Moser-Sutherland model in a sense of description of particles
with different masses. The mass ratios depend on the type of Singular orbit
stratum and are determined by its degeneracy. Using reduction due to discrete
and continuous symmetries of the system a relation to II A_n
Euler-Calogero-Moser-Sutherland model is demonstrated.Comment: 16 pages, LaTeX, no figures. V2: Typos corrected, two references
added. V3: Abstract changed, typos corrected, a few formulas and references
added. The presentation in the last section has been clarified and it was
restricted to the case of GL(3, R) group, the analysis of GL(4, R) will be
given elsewhere. V4: Minor corrections in the whole text, more formulas and
references added, accepted for publication in PL
Short Separating Geodesics for Multiply Connected Domains
We consider the following questions: given a hyperbolic plane domain and a
separation of its complement into two disjoint closed sets each of which
contains at least two points, what is the shortest closed hyperbolic geodesic
which separates these sets and is it a simple closed curve? We show that a
shortest curve always exists although in general it may not be simple. However,
one can also always find a shortest simple curve and we call such a geodesic a
\emph{meridian} of the domain. Meridians generalize to domains of higher
connectivity the notion of the equator of an annulus as the shortest geodesic
which separates the complement. We show that although they are not in general
uniquely defined, if one of the sets of the separation of the complement is
connected, then they are unique and are also the shortest possible closed
curves which separate the complement in this fashion.Comment: 20 Pages, 3 Figure
Geometric approach to non-relativistic Quantum Dynamics of mixed states
In this paper we propose a geometrization of the non-relativistic quantum
mechanics for mixed states. Our geometric approach makes use of the Uhlmann's
principal fibre bundle to describe the space of mixed states and as a novelty
tool, to define a dynamic-dependent metric tensor on the principal manifold,
such that the projection of the geodesic flow to the base manifold gives the
temporal evolution predicted by the von Neumann equation. Using that approach
we can describe every conserved quantum observable as a Killing vector field,
and provide a geometric proof for the Poincare quantum recurrence in a physical
system with finite energy levels.Comment: 19 pages, 1 figure. Minor corrections. Accepted to Journal of
Mathematical Physic
On the dynamics of the general Bianchi IX spacetime near the singularity
We show that the complex dynamics of the general Bianchi IX universe in the
vicinity of the spacelike singularity can be approximated by a simplified
system of equations. Our analysis is mainly based on numerical simulations. The
properties of the solution space can be studied by using this simplified
dynamics. Our results will be useful for the quantization of the general
Bianchi IX model.Comment: 20 pages, 5 figures, minor change
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