1,503 research outputs found
What makes nonholonomic integrators work?
A nonholonomic system is a mechanical system with velocity constraints not
originating from position constraints; rolling without slipping is the typical
example. A nonholonomic integrator is a numerical method specifically designed
for nonholonomic systems. It has been observed numerically that many
nonholonomic integrators exhibit excellent long-time behaviour when applied to
various test problems. The excellent performance is often attributed to some
underlying discrete version of the Lagrange--d'Alembert principle. Instead, in
this paper, we give evidence that reversibility is behind the observed
behaviour. Indeed, we show that many standard nonholonomic test problems have
the structure of being foliated over reversible integrable systems. As most
nonholonomic integrators preserve the foliation and the reversible structure,
near conservation of the first integrals is a consequence of reversible KAM
theory. Therefore, to fully evaluate nonholonomic integrators one has to
consider also non-reversible nonholonomic systems. To this end we construct
perturbed test problems that are integrable but no longer reversible (with
respect to the standard reversibility map). Applying various nonholonomic
integrators from the literature to these problems we observe that no method
performs well on all problems. This further indicates that reversibility is the
main mechanism behind near conservation of first integrals for nonholonomic
integrators. A list of relevant open problems is given.Comment: 27 pages, 9 figure
Geometric Hydrodynamics via Madelung Transform
We introduce a geometric framework to study Newton's equations on
infinite-dimensional configuration spaces of diffeomorphisms and smooth
probability densities. It turns out that several important PDEs of
hydrodynamical origin can be described in this framework in a natural way. In
particular, the Madelung transform between the Schr\"odinger equation and
Newton's equations is a symplectomorphism of the corresponding phase spaces.
Furthermore, the Madelung transform turns out to be a K\"ahler map when the
space of densities is equipped with the Fisher-Rao information metric. We
describe several dynamical applications of these results.Comment: 17 pages, 2 figure
Geometry of the Madelung transform
The Madelung transform is known to relate Schr\"odinger-type equations in
quantum mechanics and the Euler equations for barotropic-type fluids. We prove
that, more generally, the Madelung transform is a K\"ahler map (i.e. a
symplectomorphism and an isometry) between the space of wave functions and the
cotangent bundle to the density space equipped with the Fubini-Study metric and
the Fisher-Rao information metric, respectively. We also show that Fusca's
momentum map property of the Madelung transform is a manifestation of the
general approach via reduction for semi-direct product groups. Furthermore, the
Hasimoto transform for the binormal equation turns out to be the 1D case of the
Madelung transform, while its higher-dimensional version is related to the
problem of conservation of the Willmore energy in binormal flows.Comment: 27 pages, 2 figure
On Geodesic Completeness for Riemannian Metrics on Smooth Probability Densities
The geometric approach to optimal transport and information theory has
triggered the interpretation of probability densities as an
infinite-dimensional Riemannian manifold. The most studied Riemannian
structures are Otto's metric, yielding the -Wasserstein distance of
optimal mass transport, and the Fisher--Rao metric, predominant in the theory
of information geometry. On the space of smooth probability densities, none of
these Riemannian metrics are geodesically complete---a property desirable for
example in imaging applications. That is, the existence interval for solutions
to the geodesic flow equations cannot be extended to the whole real line. Here
we study a class of Hamilton--Jacobi-like partial differential equations
arising as geodesic flow equations for higher-order Sobolev type metrics on the
space of smooth probability densities. We give order conditions for global
existence and uniqueness, thereby providing geodesic completeness. The system
we study is an interesting example of a flow equation with loss of derivatives,
which is well-posed in the smooth category, yet non-parabolic and fully
non-linear. On a more general note, the paper establishes a link between
geometric analysis on the space of probability densities and analysis of
Euler-Arnold equations in topological hydrodynamics.Comment: 19 pages, accepted in Calc. Var. Partial Differential Equations
(2017
Diffeomorphic density matching by optimal information transport
We address the following problem: given two smooth densities on a manifold,
find an optimal diffeomorphism that transforms one density into the other. Our
framework builds on connections between the Fisher-Rao information metric on
the space of probability densities and right-invariant metrics on the
infinite-dimensional manifold of diffeomorphisms. This optimal information
transport, and modifications thereof, allows us to construct numerical
algorithms for density matching. The algorithms are inherently more efficient
than those based on optimal mass transport or diffeomorphic registration. Our
methods have applications in medical image registration, texture mapping, image
morphing, non-uniform random sampling, and mesh adaptivity. Some of these
applications are illustrated in examples.Comment: 35 page
Geometry of discrete-time spin systems
Classical Hamiltonian spin systems are continuous dynamical systems on the
symplectic phase space . In this paper we investigate the underlying
geometry of a time discretization scheme for classical Hamiltonian spin systems
called the spherical midpoint method. As it turns out, this method displays a
range of interesting geometrical features, that yield insights and sets out
general strategies for geometric time discretizations of Hamiltonian systems on
non-canonical symplectic manifolds. In particular, our study provides two new,
completely geometric proofs that the discrete-time spin systems obtained by the
spherical midpoint method preserve symplecticity.
The study follows two paths. First, we introduce an extended version of the
Hopf fibration to show that the spherical midpoint method can be seen as
originating from the classical midpoint method on for a
collective Hamiltonian. Symplecticity is then a direct, geometric consequence.
Second, we propose a new discretization scheme on Riemannian manifolds called
the Riemannian midpoint method. We determine its properties with respect to
isometries and Riemannian submersions and, as a special case, we show that the
spherical midpoint method is of this type for a non-Euclidean metric. In
combination with K\"ahler geometry, this provides another geometric proof of
symplecticity.Comment: 17 pages, 2 figures. arXiv admin note: substantial text overlap with
arXiv:1402.333
Collective symplectic integrators
We construct symplectic integrators for Lie-Poisson systems. The integrators
are standard symplectic (partitioned) Runge--Kutta methods. Their phase space
is a symplectic vector space with a Hamiltonian action with momentum map
whose range is the target Lie--Poisson manifold, and their Hamiltonian is
collective, that is, it is the target Hamiltonian pulled back by . The
method yields, for example, a symplectic midpoint rule expressed in 4 variables
for arbitrary Hamiltonians on . The method specializes in
the case that a sufficiently large symmetry group acts on the fibres of ,
and generalizes to the case that the vector space carries a bifoliation.
Examples involving many classical groups are presented
Symplectic integrators for spin systems
We present a symplectic integrator, based on the canonical midpoint rule, for
classical spin systems in which each spin is a unit vector in .
Unlike splitting methods, it is defined for all Hamiltonians, and is
-equivariant. It is a rare example of a generating function for
symplectic maps of a noncanonical phase space. It yields an integrable
discretization of the reduced motion of a free rigid body
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