22,485 research outputs found
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
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
Non-universal critical behaviour of a mixed-spin Ising model on the extended Kagome lattice
The mixed spin-1/2 and spin-3/2 Ising model on the extended Kagom\'e lattice
is solved by establishing a mapping correspondence with the eight-vertex model.
Letting the parameter of uniaxial single-ion anisotropy tend to infinity, the
model becomes exactly soluble as a free-fermion eight-vertex model. Under this
restriction, the critical points are characterized by critical exponents from
the standard Ising universality class. In a certain subspace of interaction
parameters that corresponds to a coexistence surface between two ordered
phases, the model becomes exactly soluble as a symmetric zero-field
eight-vertex model. This surface is bounded by a line of bicritical points that
have non-universal interaction-dependent critical exponents.Comment: 9 pages, 6 figure
ABJM amplitudes and the positive orthogonal grassmannian
A remarkable connection between perturbative scattering amplitudes of
four-dimensional planar SYM, and the stratification of the positive
grassmannian, was revealed in the seminal work of Arkani-Hamed et. al. Similar
extension for three-dimensional ABJM theory was proposed. Here we establish a
direct connection between planar scattering amplitudes of ABJM theory, and
singularities there of, to the stratification of the positive orthogonal
grassmannian. In particular, scattering processes are constructed through
on-shell diagrams, which are simply iterative gluing of the fundamental
four-point amplitude. Each diagram is then equivalent to the merging of
fundamental OG_2 orthogonal grassmannian to form a larger OG_k, where 2k is the
number of external particles. The invariant information that is encoded in each
diagram is precisely this stratification. This information can be easily read
off via permutation paths of the on-shell diagram, which also can be used to
derive a canonical representation of OG_k that manifests the vanishing of
consecutive minors as the singularity of all on-shell diagrams. Quite
remarkably, for the BCFW recursion representation of the tree-level amplitudes,
the on-shell diagram manifests the presence of all physical factorization
poles, as well as the cancellation of the spurious poles. After analytically
continuing the orthogonal grassmannian to split signature, we reveal that each
on-shell diagram in fact resides in the positive cell of the orthogonal
grassmannian, where all minors are positive. In this language, the amplitudes
of ABJM theory is simply an integral of a product of dlog forms, over the
positive orthogonal grassmannian.Comment: 52 pages: v2, typos corrected, published version in JHE
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