13,256 research outputs found

### The free rigid body dynamics: generalized versus classic

In this paper we analyze the normal forms of a general quadratic Hamiltonian
system defined on the dual of the Lie algebra $\mathfrak{o}(K)$ of real $K$ -
skew - symmetric matrices, where $K$ is an arbitrary $3\times 3$ real symmetric
matrix. A consequence of the main results is that any first-order autonomous
three-dimensional differential equation possessing two independent quadratic
constants of motion which admits a positive/negative definite linear
combination, is affinely equivalent to the classical "relaxed" free rigid body
dynamics with linear controls.Comment: 12 page

### How to mesh up Ewald sums (I): A theoretical and numerical comparison of various particle mesh routines

Standard Ewald sums, which calculate e.g. the electrostatic energy or the
force in periodically closed systems of charged particles, can be efficiently
speeded up by the use of the Fast Fourier Transformation (FFT). In this article
we investigate three algorithms for the FFT-accelerated Ewald sum, which
attracted a widespread attention, namely, the so-called
particle-particle-particle-mesh (P3M), particle mesh Ewald (PME) and smooth PME
method. We present a unified view of the underlying techniques and the various
ingredients which comprise those routines. Additionally, we offer detailed
accuracy measurements, which shed some light on the influence of several tuning
parameters and also show that the existing methods -- although similar in
spirit -- exhibit remarkable differences in accuracy. We propose combinations
of the individual components, mostly relying on the P3M approach, which we
regard as most flexible.Comment: 18 pages, 8 figures included, revtex styl

### How Close to Two Dimensions Does a Lennard-Jones System Need to Be to Produce a Hexatic Phase?

We report on a computer simulation study of a Lennard-Jones liquid confined
in a narrow slit pore with tunable attractive walls. In order to investigate
how freezing in this system occurs, we perform an analysis using different
order parameters. Although some of the parameters indicate that the system goes
through a hexatic phase, other parameters do not. This shows that to be certain
whether a system has a hexatic phase, one needs to study not only a large
system, but also several order parameters to check all necessary properties. We
find that the Binder cumulant is the most reliable one to prove the existence
of a hexatic phase. We observe an intermediate hexatic phase only in a
monolayer of particles confined such that the fluctuations in the positions
perpendicular to the walls are less then 0.15 particle diameters, i. e. if the
system is practically perfectly 2d

### Complete integrability versus symmetry

The purpose of this article is to show that on an open and dense set,
complete integrability implies the existence of symmetry

### Continuous and discrete Clebsch variational principles

The Clebsch method provides a unifying approach for deriving variational
principles for continuous and discrete dynamical systems where elements of a
vector space are used to control dynamics on the cotangent bundle of a Lie
group \emph{via} a velocity map. This paper proves a reduction theorem which
states that the canonical variables on the Lie group can be eliminated, if and
only if the velocity map is a Lie algebra action, thereby producing the
Euler-Poincar\'e (EP) equation for the vector space variables. In this case,
the map from the canonical variables on the Lie group to the vector space is
the standard momentum map defined using the diamond operator. We apply the
Clebsch method in examples of the rotating rigid body and the incompressible
Euler equations. Along the way, we explain how singular solutions of the EP
equation for the diffeomorphism group (EPDiff) arise as momentum maps in the
Clebsch approach. In the case of finite dimensional Lie groups, the Clebsch
variational principle is discretised to produce a variational integrator for
the dynamical system. We obtain a discrete map from which the variables on the
cotangent bundle of a Lie group may be eliminated to produce a discrete EP
equation for elements of the vector space. We give an integrator for the
rotating rigid body as an example. We also briefly discuss how to discretise
infinite-dimensional Clebsch systems, so as to produce conservative numerical
methods for fluid dynamics

### Lagrangian Reduction, the Euler--Poincar\'{e} Equations, and Semidirect Products

There is a well developed and useful theory of Hamiltonian reduction for
semidirect products, which applies to examples such as the heavy top,
compressible fluids and MHD, which are governed by Lie-Poisson type equations.
In this paper we study the Lagrangian analogue of this process and link it with
the general theory of Lagrangian reduction; that is the reduction of
variational principles. These reduced variational principles are interesting in
their own right since they involve constraints on the allowed variations,
analogous to what one finds in the theory of nonholonomic systems with the
Lagrange d'Alembert principle. In addition, the abstract theorems about
circulation, what we call the Kelvin-Noether theorem, are given.Comment: To appear in the AMS Arnold Volume II, LATeX2e 30 pages, no figure

### A Note from the Editor

A note from the editor of Speaker & Gavel, Todd Holm for volume 58, issue 1, 2022

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