356 research outputs found
The angular momentum of a relative equilibrium
There are two main reasons why relative equilibria of N point masses under
the influence of Newton attraction are mathematically more interesting to study
when space dimension is at least 4: On the one hand, in a higher dimensional
space, a relative equilibrium is determined not only by the initial
configuration but also by the choice of a complex structure on the space where
the motion takes place; in particular, its angular momentum depends on this
choice; On the other hand, relative equilibria are not necessarily periodic: if
the configuration is "balanced" but not central, the motion is in general
quasi-periodic. In this exploratory paper we address the following question,
which touches both aspects: what are the possible frequencies of the angular
momentum of a given central (or balanced) configuration and at what values of
these frequencies bifurcations from periodic to quasi-periodic relative
equilibria do occur ? We give a full answer for relative equilibrium motions in
dimension 4 and conjecture that an analogous situation holds true for higher
dimensions. A refinement of Horn's problem given by Fomin, Fulton, Li and Poon
plays an important role.
P.S. The conjecture is now proved (see Alain Chenciner and Hugo Jimenez
Perez, Angular momentum and Horn's problem, arXiv:1110.5030v1 [math.DS]).Comment: 17 pages, 3 figure
Non-avoided crossings for n-body balanced configurations in R^3 near a central configuration
The balanced configurations are those n-body configurations which admit a
relative equilibrium motion in a Euclidean space E of high enough dimension 2p.
They are characterized by the commutation of two symmetric endomorphisms of the
(n-1)-dimensional Euclidean space of codispositions, the intrinsic inertia
endomorphism B which encodes the shape and the Wintner-Conley endomorphism A
which encodes the forces. In general, p is the dimension d of the
configuration, which is also the rank of B. Lowering to 2(d-1) the dimension of
E occurs when the restriction of A to the (invariant) image of B possesses a
double eigenvalue. It is shown that, while in the space of all dxd-symmetric
endomorphisms, having a double eigenvalue is a condition of codimension 2 (the
avoided crossings of physicists), here it becomes of codimension 1 provided
some condition (H) is satisfied. As the condition is always satisfied for
configurations of the maximal dimension (i.e. if d=n-1), this implies in
particular the existence, in the neighborhood of the regular tetrahedron
configuration of 4 bodies with no three of the masses equal, of exactly 3
families of balanced configurations which admit relative equilibrium motion in
a four dimensional space.Comment: 35 pages, 1 diagram, 6 figures Section 1.5.2 is new: it introduces
the condition (H) which had been overlooked in the first versio
Between two moments
In this short note, we draw attention to a relation between two Horn
polytopes which is proved in [Chenciner-Jim\'enez P\'erez] as the result on the
one side of a deep combinatorial result in [Fomin,Fulton, Li,Poon], on the
other side of a simple computation involving complex structures. This suggested
an inequality between Littlewood-Richardson coefficients which we prove using
the symmetric characterization of these coefficients given in
[Carr\'e,Leclerc].Comment: 9 pages, 3 figure
A remarkable periodic solution of the three-body problem in the case of equal masses
Using a variational method, we exhibit a surprisingly simple periodic orbit
for the newtonian problem of three equal masses in the plane. The orbit has
zero angular momentum and a very rich symmetry pattern. Its most surprising
feature is that the three bodies chase each other around a fixed eight-shaped
curve. Setting aside collinear motions, the only other known motion along a
fixed curve in the inertial plane is the ``Lagrange relative equilibrium" in
which the three bodies form a rigid equilateral triangle which rotates at
constant angular velocity within its circumscribing circle. Our orbit visits in
turns every ``Euler configuration" in which one of the bodies sits at the
midpoint of the segment defined by the other two (Figure 1). Numerical
computations by Carles Sim\'o, to be published elsewhere, indicate that the
orbit is ``stable" (i.e. completely elliptic with torsion). Moreover, they show
that the moment of inertia I(t) with respect to the center of mass and the
potential U(t) as functions of time are almost constant.Comment: 21 pages, published versio
Periodic orbits for three particles with finite angular momentum
We present a novel numerical method to calculate periodic orbits for
dynamical systems by an iterative process which is based directly on the action
integral in classical mechanics. New solutions are obtained for the planar
motion of three equal mass particles on a common periodic orbit with finite
total angular momentum, under the action of attractive pairwise forces of the
form . It is shown that for , Lagrange's 1772 circular
solution is the limiting case of a complex symmetric orbit. The evolution of
this orbit and another recently discovered one in the shape of a figure eight
is investigated for a range of angular momenta. Extensions to n equal mass
particles and to three particles of different masses are also discussed
briefly.Comment: 6 pages, 7 figures. Accepted for publication in Physics Letters
Action minimizing solutions of the Newtonian n-body problem: from homology to symmetry
An action minimizing path between two given configurations, spatial or
planar, of the -body problem is always a true -- collision-free -- solution.
Based on a remarkable idea of Christian Marchal, this theorem implies the
existence of new "simple" symmetric periodic solutions, among which the Eight
for 3 bodies, the Hip-Hop for 4 bodies and their generalizations
Action minimizing orbits in the n-body problem with simple choreography constraint
In 1999 Chenciner and Montgomery found a remarkably simple choreographic
motion for the planar 3-body problem (see \cite{CM}). In this solution 3 equal
masses travel on a eight shaped planar curve; this orbit is obtained minimizing
the action integral on the set of simple planar choreographies with some
special symmetry constraints. In this work our aim is to study the problem of
masses moving in \RR^d under an attractive force generated by a potential
of the kind , , with the only constraint to be a simple
choreography: if are the orbits then we impose the
existence of x \in H^1_{2 \pi}(\RR,\RR^d) such that q_i(t)=x(t+(i-1) \tau),
i=1,...,n, t \in \RR, where . In this setting, we first
prove that for every d,n \in \NN and , the lagrangian action
attains its absolute minimum on the planar circle. Next we deal with the
problem in a rotating frame and we show a reacher phenomenology: indeed while
for some values of the angular velocity minimizers are still circles, for
others the minima of the action are not anymore rigid motions.Comment: 24 pages; 4 figures; submitted to Nonlinearit
Saari's homographic conjecture for planar equal-mass three-body problem under a strong force potential
Donald Saari conjectured that the -body motion with constant
configurational measure is a motion with fixed shape. Here, the configurational
measure is a scale invariant product of the moment of inertia and the potential function , . Namely, . We will show
that this conjecture is true for planar equal-mass three-body problem under the
strong force potential
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