The spatial three-body problem admits the ten integrals of energy, linear momentum and angular momentum. Fixing these integrals defines an eight dimensional algebraic set called the integral manifold, M(c,h), which depends on the energy level h and the magnitude c of the angular momentum vector. The seven dimensional reduced integral manifold, MR(c,h), is the quotient space M(c,h)/SO2 where the SO2 action is rotation about the angular momentum vector. We want to determine how the topology of M(c,h) and MR(c,h) depends on c and h. It turns out that there is one bifurcation parameter, n = -c2 h, and nine special values of this parameter, ni, i=1,...,9. At each of the special values the geometric restrictions imposed by the integrals change, but one of these values, n5, does not give rise to a change in the topology of the integral manifolds M(c,h) and MR(c,h). The other eight special values give rise to nine different topologically distinct cases. We give a complete description of the geometry of these sets along with their homology. These results confirm some conjectures and refutes several others. In particular Birkhoff's conjecture that bifurcations occur only at the four central configurations is shown to be false in the spatial problem.
We not only study the integral manifolds, but also the six dimensional
Hill's regions, H(c,h), and the five dimensional reduced Hill's region
HR(c,h) = H(c,h)/SO2. The Hill's region is the projection of
the integral manifold onto position space. We determine the homotopy type
of the reduced Hill's region, and the homology groups of H(c,h) and HR(c,h).
This does not detect all the changes in the homeomorphism type of H(c,h),
so we investigate the topology of the boundary points of these Hill's regions.
From this finer analysis, we conclude that these Hill's regions undergo
bifurcations at eight of the special values of the bifurcation parameter,
but none of the topological invariants that we calculate detect a bifurcation
at the parameter value n5.
One of the questions left open in the three-body
memoir was whether or not the topology of the Hill's region changes
at the parameter value n5. In this paper, it is shown
that it does not: the topology of the Hill's region changes if and
only if the topology of the integral manifold changes.
In Dynamical Systems, Birkhoff gave a clear formulation of a
cross section, suggested a possible generalization to a cross section with
boundary, and raised the question of whether or not such cross sections
exist in the three-body problem. In this work, we explicitly develop
Birkhoff's notion of a generalized cross section, formulate necessary conditions
for the existence of a cross section or generalized cross section, and
show that these conditions are not satisfied in the three-body problem.
The homological conditions are similar to those of the Zeta
functions paper.
We compute the homology of the integral manifolds of the restrictedthree-body problem --- planar and spatial, unregularized and regularized. Holding the Jacobi constant fixed defines a three dimensional algebraic set in the planar case and a five dimensional algebraic set in the spatial case (the integral manifolds). The singularities of the restricted problem due to collusions are removable which defines the regularized problem.
There are five positive critical values of the Jacobi constant: one is due to a critical point at infinity, another is due to the Lagrangian critical points, and three are due to the Eulerian critical points. The critical point at infinity occurs only in the spatial problems. We compute the homology of the integral manifold for each regular value of the Jacobi constant. These computations show that at each critical value the integral manifolds undergo a bifurcation in their topology. The bifurcation due to a critical point at infinity shows that Birkhoff's conjecture is false even in the restricted problem.
Birkhoff also asked if the planar problem is the boundary of a cross section for the spatial problem. Our computations and homological criteria show that this can never happen in the restricted problem, but may be possible in the regularized problem for some values of the Jacobi constant. We also investigate the existence of global cross sections in each of the problems.
In the planar N-body problem, N point masses move in the plane under their mutual gravitational attraction. It is classical that the dynamics of this motion conserves the integrals of motion: center of mass, linear momentum, angular momentum c, and energy h. Further, the motion has a rotational symmetry. The dynamics thus takes place on a (4N-7)-dimensional open manifold, known as the reduced integral manifold mR(M,n). The topology of this manifold depends only on the masses M = (m1, ..., mN) and the quantity n = h|c|2.
In spite of the central importance of this manifold
in a classical dynamical problem, very little is known about the topology of
mR(M,n).
In this note, we build on the topological analysis of Smale to describe the
homology of mR(M,n).
A variety of homological results are presented, including the computation of
the homology groups for n
very large for all M; and for all n
for three masses, and for four equal masses.
The integral manifolds of the N-body problem are the level sets of energy
and angular momentum. For positive energy and non-zero angular momentum, all
level sets are diffeomorphic to a non-zero level set of angular momentum on
the unit tangent bundle of the configuration space. The one complication
that arises in attempting to describe this level set explicitly
is the degeneracy at the syzygies of the equations that define angular momentum.
In this work, we analyze the behavior of the angular momentum near syzygies,
and show how to construct local coordinates near the syzygies. In particular,
we show that the projection of the integral manifold onto the configuration
space Kc is a homotopy equivalence, and use this to compute the
homology of the integral manifolds.
In the N-body problem, it is a simple observation that relative equilibria (planar solutions for which the mutual distances between the particles remain constant) have constant moment of inertia. That is, trajectories with constant shape have constant size. In 1970, Don Saari conjectured that the converse was true: if a solution to the N-body problem has constant moment of inertia, then it must be a relative equilibrium. In this note, we confirm the conjecture for N = 3 with equal masses.
For all masses, there are at least n-2 O2-orbits of non-collinear
planar central configurations. In particular, this estimate is valid even
if the potential function is not a Morse function. If the potential function
is a Morse function, then an improved lower bound, on the order of (n!/2)ln((n+1)/3),
can be given.
We consider the problem of N equally charged particles, confined to
the surface of a sphere and moving under their mutual electrostatic forces.
This system is of interest physically, and also displays interesting similarities
witht the N-body problem of celestial mechanics. However, the dynamics
of this system are not well-understood. In this paper, we address one of
the basic dynamical questions: counting, locating, and determining the
stability of the rest points of the system. These rest points correspond
to distributions of the charges on the surface of the sphere in which the
total repulsive force is directed radially.
A nilmanifold is a homogenous space of a nilpotent Lie group. If M is a compact symplectic nilmanifold, then any 1-periodic Hamiltonian system on M has at least dim(M) + 1 contractible periodic orbits with period 1. This provides an affirmative answer to the Arnold conjecture for such manifolds. The proof uses the techniques of rational homotopy theory, and in particular an extension of the rational L.-S. category invariant e0 to maps.
Given an unknown attractor A in a continuous dynamical system,
how can we discover the topology and dynamics of A? As a practical
matter, how can we do so from only a finite amount of information?
One way of doing so is to produce a semi-conjugacy from A onto a
model system M whose topology and dynamics are known. The
complexity of M then provides a lower bound for the complexity of
A. This paper uses the techniques of the Conley index to construct
a simplicial model and a surjective semi-conjugacy for a large class of
attractors. The essential features of this construction is that the
model M can be explicitly described; and that the finite amount
of information needed to construct it is computable.
This paper is an introduction to connection and transition matrices in the Conley index theory for flows. Basic definitions and simple examples are discussed.
This is an expository summary of the simplicial modeling of global attractors described in Simplicial models for the global dynamics of attractors.
A semi-conjugacy from the dynamics of the global attractors for a family
of scalar delay differential equations with negative feedback onto
the dynamics of a simple system of ordinary differential equations is constructed.
The construction and proof are done in an abstract setting, and hence,
are valid a variety of dynamical systems which need not arise from delay
equations. The proofs are based on the Conley index theory.
The topological transition matrices introduced
by Konstantin Mischaikow and I were so named because they were similar
in form to the (singular) transition matrices defined by Jim Reineck.
In this paper, we show that our topological transition matrices are indeed
special cases of Jim's singular transition matrices. In the process,
we clarify some of the foundational issues for connection matrices and
transition matrices.
We use the Conley index to produce a simple sufficient condition (stated
in terms of the Betti numbers of the cohomology Conley index) for the existence
of periodic orbits for flows which admit a cross-section.
Given invariant sets A, B and C, and connecting orbits from A to B and
from B to C, we state very general conditions under which they bifurcate
to produce an connecting orbit from A to C. In particular, our theorem
is applicable in settings for which one has an admissible semiflow on an
isolating neighborhood of the invariant sets and connecting orbits, and
for which the Conley index of the invariant sets is the same as that of
a hyperbolic critical point. Our proof depends on the connected simple
system associated with the Conley index for isolated invariant sets. Furthermore,
we show how this change in connected simple systems can be associated with
transition matrices, and hence, connection matrices. This leads to some
simple examples in which the nonuniqueness of the connection matrix can
be explained by changes in the connected simple system.
The Conley index for continuous dynamical systems is defined for (one-sided)
semiflows. For (two-sided) flows, there are two indices defined: one for
the forward flow; and one for the reverse flow. In general, the two indices
give different information about the flow; but for flows on orientable
manifolds, there is a duality isomorphism between the homology Conley indices
of the forward and reverse flows. This duality preserves the algebraic
structure of many of the constructions of the Conley index theory: sums
and products; continuation; attractor-repeller sequences and connection
matrices.
The following generalization of the Poincare-Hopf index theorem is proved:
If S is an isolated invariant set of a flow on a manifold M, then the sum
of the Hopf indices on S is equal (up to a sign) to the Euler characteristic
of the homology Conley index of S.
In the Conley index theory, the connection map of the homology attractor-repller
sequence provides a means of detecting connecting orbits between a repeller
and attractor in an isolated invariant set. In this work, the connection
map is shown to be additive: under suitable decompositions of the connecting
orbit set, the connection map of the invariant set equals the sum of the
connection maps of the decomposition elements. This refines the information
provided by the homology attractor-repeller sequence. In particular, the
properties of the connection map lead to a characterization of isolated
invariant sets with hyperbolic critical points as an attractor-repeller
pair.
In addition to the many different Nielsen-type numbers that have been
introduced to study fixed points, there are Nielsen-type numbers that have
been created to study coincidences of maps; intersections
of maps; and preimages of maps. These problems, while distinct,
are all similar, and the Nielsen theories that have been created to study
them display strong structural similarities as well. This paper explores
those similarities, ans shows that the relations between the three theories
are closer and more formal than just similarity. There are transformations
that allow any of the three Nielsen problems to be converted into either
of the other two. Analysis of these transformations allows us to
make precise the relationships between the three Nielsen theories.
Nielsen theory, originally developed as a homotopy-theoretic approach to fixed point theory, has been translated and extended to various other problems, such as the study of periodic points, coincidence points and roots. Recently, the techniques of Nielsen theory have been applied to the study of intersections of maps. A Nielsen- type number, the Nielsen intersection number NI(f,g), was introduced, and shown to have many of the properties analogous to those of the Nielsen fixed point number. In particular, it is a homotopy-invariant lower bound for the number of intersections of a pair of maps. The question of whether or not this lower bound is sharp can be thought of as the Wecken problem for intersection theory. In this paper, the Wecken problem for intersections is considered, and some Wecken theorems are proved.
Nielsen theory, originally developed as a homotopy-theoretic approach
to fixed point theory, has been translated and extended to various other
problems, such as the study of periodic points, coincidence points and
roots. In this paper, the techniques of Nielsen theory are applied to the
study of intersections of pairs of maps f:X "
Z ! Y:g. A Nielsen-type
number, the Nielsen intersection number NI(f,g), is introduced, and shown
to have many of the properties analogous to those of the Nielsen fixed
point number. In particular, NI(f,g) gives a lower bound for the number
of points of intersection for all maps homotopic to f and g.
In Lefschetz and Nielsen coincidence numbers on solvmanifolds,
it was claimed that Nielsen coincidence numbers and Lefschetz coincidence
numbers are related by the inequality N(f,g) > |L(f,g)| for all maps f,g:S1
" S2 between compact orientable solvmanifolds of the same
dimension. It was further claimed that N(f,g) = |L(f,g)| when S2
is a nilmanifold. A mistake in that paper has been discovered. In this
paper, that mistake is partially repaired. A new proof of the equality
N(f,g) = |L(f,g)| for nilmanifolds is given, and a variety of conditions
for maps on orientable solvmanifolds are established which imply the inequality
N(f,g) > |L(f,g)|. However, it still remains open whether N(f,g) >
|L(f,g)|
for all maps between orientable solvmanifolds.
A well-known lower bound for the number of fixed points of a self map
f:X " X is the Nielsen number N(f). Unfortunately, the Nielsen number
is difficult to calculate. The Lefschetz number L(f), on the other hand,
is readily computable, but usually does not estimate the number of fixed
points. It is known that N(f) = |L(f)| for all maps on nilmanifolds (homogeneous
spaces of nilpotent Lie groups) and that N(f) > |L(f)| for all maps on
solvmanifolds (homogeneous spaces of solvable Lie groups). Typically, though,
the strict inequality holds, so the Nielsen number cannot be completely
computed from the Lefschetz number. In the present work, we produce a large
class of solvmanifolds for which N(f) = |L(f)| for all self maps. This
class includes exponential solvmanifolds: solvmanifolds for which the corresponding
exponential map is surjective. Our methods provide Nielsen and Lefschtez
number product theorems for the Mostow fibrations of these solvmanifolds
even though the maps on the fibers in general will belong to varying homotopy
classes.
A well-known lower bound for the number of fixed points of a self-map
f:X " X is the Nielsen number N(f). Unfortunately, the Nielsen number
is difficult to calculate. The Lefschetz number L(f), on the other hand,
is readily computable, but usually does not estimate the number of fixed
points. In this paper, we show that on infrasolvmanifolds (aspherical manifolds
whose fundamental group has a normal solvable group of finite index), N(f)
= L(f) when f is a homotopically periodic map.
For Nielsen numbers to be truly useful in applied problems, they should
be computable. Since it is generally impossible to compute Nielsen numbers
directly from the definition, other methods of computation must be developed.
In this paper, some of the existing computational methods are surveyed.
All of these methods rely on a combination of homological and fundamental
group information for their application.
A well-known lower bound for the number of fixed points of a self-map
f:X " X is the Nielsen number N(f). Unfortunately, the Nielsen number
is difficult to calculate. The Lefschetz number L(f), on the other hand,
is readily computable, but does not give a lower bound for the number of
fixed points. In this paper, we investigate conditions on the space X which
guarantee either N(f) = |L(f)| or N(f) > |L(f)|. By considering the Nielsen
and Lefschetz coincidence numbers, we show that N(f) > |L(f)| for all
self-maps on compact infrasolvmanifolds (aspherical manifolds whose fundamental
group has a normal solvable group of finite index). Moreover, for infranilmanifolds,
there is a Lefschetz number formula which computes N(f).
Suppose M1, M2 are compact, connected orientable manifolds of the same dimension. Then for all pairs of maps f,g:M1 " M2, the Nielsen coincidence number N(f,g) and the Lefschetz coincidence number L(f,g) are measures of the number of coincidences of f and g: points x in M1 with f(x) = g(x). A manifold is a nilmanifold (solvmanifold) if it is a homogeneous space of a nilpotent (solvable) Lie group. If M1 and M2 are compact connected orientable solvmanifolds, then N(f, g) > |L(f, g)| for all f and g, with equality for all f and g if M2 is a nilmanifold.
There is an error in this paper. See Lefschetz
and Nielsen coincidence numbers on nilmanifolds and solvmanifolds, II.
A compact solvmanifold S is a homogeneous space of a simply connected solvable Lie group: S = G/H, with H a uniform subgroup of G. If f:S " S is a continuous self-map on S, we show that N(f) > |L(f)|, where N(f) is the Nielsen number of f and L(f) is the Lefschetz number. Necessary and sufficient conditions, stated in terms of the fundamental group, are given for the equality N(f) = |L(f)| to hold.
The two-dimensional cutting stock problem is the problem of cutting two-dimensional parts from a sheet such that the cutting requirements are met and no cuts overlap.
This paper presents a genetic algorithm for the cutting-stock problem which handles non-convex, irregularly shaped parts. The objective is to maximize the utilization of the sheet for a given set of parts that need to be cut from it. The salient features of this algorithm are the way in which layouts are described, and the way in which the efficiency function is computed. We create ‘pseudo-layouts’ of the parts on the sheet , which are easier to be manipulated by the genetic algorithm. These pseudo layouts can be changed into unique actual layouts in a simple manner.
The proposed algorithm can handle placement of highly irregular parts with ease as no simplifying assumptions are made about the shape of the parts to be placed. The algorithm was tested with part requirements resembling those from an actual leather industry stock cut need. Several runs were made with different part requirements and varying sheet dimensions. The efficiency of placement ranged from 65-75%.
The two-dimensional stock cutting problem is a well-known and oft-studied problem. We have developed a genetic algorithm approach to the problem, which is capable of handling some of the more intractable forms of the problem: non-convex parts; non-convex sheets; multiple irregularly shaped sheets; etc. We are developing an integrated manufacturing system, which incorporates a machine vision module for acquisition of the images of irregular (non-convex) parts and sheets, polygonalizing them and storing them in a database of parts and sheets. Using the polygonal images as well as the manufacturing schedules and priorities as input, genetic algorithms are used to generate part layouts that satisfy the manufacturing constraints. The significant features of this approach are the integration of all aspects of the layout process; and the flexibility of the genetic algorithm approach, that allows it to be adapted to fit the special requirements of different problems. The proposed methods can be particularly useful in leather and apparel industries where non-convex parts and sheets are commonly used.