47 research outputs found
The Kepler Problem with Anisotropic Perturbations
We study a 2-body problem given by the sum of the Newtonian potential and an
anisotropic perturbation that is a homogeneous function of degree ,
. For , the sets of initial conditions leading to
collisions/ejections and the one leading to escapes/captures have positive
measure. For and , the flow on the zero-energy manifold
is chaotic. For , a case we prove integrable, the infinity manifold of
the zero-energy level is a disconnected set, which has heteroclinic connections
with the collision manifold
Classical dynamics near the triple collision in a three-body Coulomb problem
We investigate the classical motion of three charged particles with both
attractive and repulsive interaction.The triple collision is a main source of
chaos in such three body Coulomb problems.By employing the McGehee scaling
technique, we analyse here for the first time in detail the three-body dynamics
near the triple collision in 3 degrees of freedom.We reveal surprisingly simple
dynamical patterns in large parts of the chaotic phase space. The underlying
degree of order in the form of approximate Markov partitions may help in
understanding the global structures observed in quantum spectra of two-electron
atoms.Comment: 4 pages, 3 figure
Symmetric Periodic Solutions of the Anisotropic Manev Problem
We consider the Manev Potential in an anisotropic space, i.e., such that the
force acts differently in each direction. Using a generalization of the
Poincare' continuation method we study the existence of periodic solutions
for weak anisotropy. In particular we find that the symmetric periodic orbits
of the Manev system are perturbed to periodic orbits in the anisotropic
problem.Comment: Late
The classical dynamics of two-electron atoms near the triple collision
The classical dynamics of two electrons in the Coulomb potential of an
attractive nucleus is chaotic in large parts of the high-dimensional phase
space. Quantum spectra of two-electron atoms, however, exhibit structures which
clearly hint at the existence of approximate symmetries in this system. In a
recent paper,(Phys. Rev. Lett. 93, 054302 (2004)), we presented a study of the
dynamics near the triple collision as a first step towards uncovering the
hidden regularity in the classical dynamics of two electron atoms. The
non-regularisable triple collision singularity is a main source of chaos in
three body Coulomb problems. Here, we will give a more detailed account of our
findings based on a study of the global structure of the stable and unstable
manifolds of the triple collision.Comment: 21 pages, 17 figure
Background-Independence
Intuitively speaking, a classical field theory is background-independent if
the structure required to make sense of its equations is itself subject to
dynamical evolution, rather than being imposed ab initio. The aim of this paper
is to provide an explication of this intuitive notion. Background-independence
is not a not formal property of theories: the question whether a theory is
background-independent depends upon how the theory is interpreted. Under the
approach proposed here, a theory is fully background-independent relative to an
interpretation if each physical possibility corresponds to a distinct spacetime
geometry; and it falls short of full background-independence to the extent that
this condition fails.Comment: Forthcoming in General Relativity and Gravitatio
The use of normal forms for analysing nonlinear mechanical vibrations.
A historical introduction is given of the theory of normal forms for simplifying nonlinear dynamical systems close to resonances or bifurcation points. The specific focus is on mechanical vibration problems, described by finite degree-of-freedom second-order-in-time differential equations. A recent variant of the normal form method, that respects the specific structure of such models, is recalled. It is shown how this method can be placed within the context of the general theory of normal forms provided the damping and forcing terms are treated as unfolding parameters. The approach is contrasted to the alternative theory of nonlinear normal modes (NNMs) which is argued to be problematic in the presence of damping. The efficacy of the normal form method is illustrated on a model of the vibration of a taut cable, which is geometrically nonlinear. It is shown how the method is able to accurately predict NNM shapes and their bifurcations
Relational Particle Models. II. Use as toy models for quantum geometrodynamics
Relational particle models are employed as toy models for the study of the
Problem of Time in quantum geometrodynamics. These models' analogue of the thin
sandwich is resolved. It is argued that the relative configuration space and
shape space of these models are close analogues from various perspectives of
superspace and conformal superspace respectively. The geometry of these spaces
and quantization thereupon is presented. A quantity that is frozen in the scale
invariant relational particle model is demonstrated to be an internal time in a
certain portion of the relational particle reformulation of Newtonian
mechanics. The semiclassical approach for these models is studied as an
emergent time resolution for these models, as are consistent records
approaches.Comment: Replaced with published version. Minor changes only; 1 reference
correcte
Motion in classical field theories and the foundations of the self-force problem
This article serves as a pedagogical introduction to the problem of motion in
classical field theories. The primary focus is on self-interaction: How does an
object's own field affect its motion? General laws governing the self-force and
self-torque are derived using simple, non-perturbative arguments. The relevant
concepts are developed gradually by considering motion in a series of
increasingly complicated theories. Newtonian gravity is discussed first, then
Klein-Gordon theory, electromagnetism, and finally general relativity. Linear
and angular momenta as well as centers of mass are defined in each of these
cases. Multipole expansions for the force and torque are then derived to all
orders for arbitrarily self-interacting extended objects. These expansions are
found to be structurally identical to the laws of motion satisfied by extended
test bodies, except that all relevant fields are replaced by effective versions
which exclude the self-fields in a particular sense. Regularization methods
traditionally associated with self-interacting point particles arise as
straightforward perturbative limits of these (more fundamental) results.
Additionally, generic mechanisms are discussed which dynamically shift ---
i.e., renormalize --- the apparent multipole moments associated with
self-interacting extended bodies. Although this is primarily a synthesis of
earlier work, several new results and interpretations are included as well.Comment: 68 pages, 1 figur