141 research outputs found
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
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
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
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
Choreographic Three Bodies on the Lemniscate
We show that choreographic three bodies {x(t), x(t+T/3), x(t-T/3)} of period
T on the lemniscate, x(t) = (x-hat+y-hat cn(t))sn(t)/(1+cn^2(t)) parameterized
by the Jacobi's elliptic functions sn and cn with modulus k^2 = (2+sqrt{3})/4,
conserve the center of mass and the angular momentum, where x-hat and y-hat are
the orthogonal unit vectors defining the plane of the motion. They also
conserve the moment of inertia, the kinetic energy, the sum of square of the
curvature, the product of distance and the sum of square of distance between
bodies. We find that they satisfy the equation of motion under the potential
energy sum_{i<j}(1/2 ln r_{ij} -sqrt{3}/24 r_{ij}^2) or sum_{i<j}1/2 ln r_{ij}
-sum_{i}sqrt{3}/8 r_{i}^2, where r_{ij} the distance between the body i and j,
and r_{i} the distance from the origin. The first term of the potential
energies is the Newton's gravity in two dimensions but the second term is the
mutual repulsive force or a repulsive force from the origin, respectively.
Then, geometric construction methods for the positions of the choreographic
three bodies are given
Simple choreographies of the planar Newtonian -body Problem
In the -body problem, a simple choreography is a periodic solution, where
all masses chase each other on a single loop. In this paper we prove that for
the planar Newtonian -body problem with equal masses, , there are
at least different main simple choreographies. This
confirms a conjecture given by Chenciner and etc. in \cite{CGMS02}.Comment: 31pages, 6 figures. Refinements in notations and proof
Kustaanheimo-Stiefel Regularization and the Quadrupolar Conjugacy
In this note, we present the Kustaanheimo-Stiefel regularization in a
symplectic and quaternionic fashion. The bilinear relation is associated with
the moment map of the - action of the Kustaanheimo-Stiefel
transformation, which yields a concise proof of the symplecticity of the
Kustaanheimo-Stiefel transformation symplectically reduced by this circle
action. The relation between the Kustaanheimo-Stiefel regularization and the
Levi-Civita regularization is established via the investigation of the
Levi-Civita planes. A set of Darboux coordinates (which we call
Chenciner-F\'ejoz coordinates) is generalized from the planar case to the
spatial case. Finally, we obtain a conjugacy relation between the integrable
approximating dynamics of the lunar spatial three-body problem and its
regularized counterpart, similar to the conjugacy relation between the extended
averaged system and the averaged regularized system in the planar case.Comment: 19 pages, corrected versio
Straight Line Orbits in Hamiltonian Flows
We investigate periodic straight-line orbits (SLO) in Hamiltonian force
fields using both direct and inverse methods. A general theorem is proven for
natural Hamiltonians quadratic in the momenta in arbitrary dimension and
specialized to two and three dimension. Next we specialize to homogeneous
potentials and their superpositions, including the familiar H\'enon-Heiles
problem. It is shown that SLO's can exist for arbitrary finite superpositions
of -forms. The results are applied to a family of generalized H\'enon-Heiles
potentials having discrete rotational symmetry. SLO's are also found for
superpositions of these potentials.Comment: laTeX with 6 figure
An Exactly Conservative Integrator for the n-Body Problem
The two-dimensional n-body problem of classical mechanics is a non-integrable
Hamiltonian system for n > 2. Traditional numerical integration algorithms,
which are polynomials in the time step, typically lead to systematic drifts in
the computed value of the total energy and angular momentum. Even symplectic
integration schemes exactly conserve only an approximate Hamiltonian. We
present an algorithm that conserves the true Hamiltonian and the total angular
momentum to machine precision. It is derived by applying conventional
discretizations in a new space obtained by transformation of the dependent
variables. We develop the method first for the restricted circular three-body
problem, then for the general two-dimensional three-body problem, and finally
for the planar n-body problem. Jacobi coordinates are used to reduce the
two-dimensional n-body problem to an (n-1)-body problem that incorporates the
constant linear momentum and center of mass constraints. For a four-body
choreography, we find that a larger time step can be used with our conservative
algorithm than with symplectic and conventional integrators.Comment: 17 pages, 3 figures; to appear in J. Phys. A.: Math. Ge
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