6 research outputs found
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
Saari's homographic conjecture for planar equal-mass three-body problem in Newton gravity
Saari's homographic conjecture in N-body problem under the Newton gravity is
the following; configurational measure \mu=\sqrt{I}U, which is the product of
square root of the moment of inertia I=(\sum m_k)^{-1}\sum m_i m_j r_{ij}^2 and
the potential function U=\sum m_i m_j/r_{ij}, is constant if and only if the
motion is homographic. Where m_k represents mass of body k and r_{ij}
represents distance between bodies i and j. We prove this conjecture for planar
equal-mass three-body problem.
In this work, we use three sets of shape variables. In the first step, we use
\zeta=3q_3/(2(q_2-q_1)) where q_k \in \mathbb{C} represents position of body k.
Using r_1=r_{23}/r_{12} and r_2=r_{31}/r_{12} in intermediate step, we finally
use \mu itself and \rho=I^{3/2}/(r_{12}r_{23}r_{31}). The shape variables \mu
and \rho make our proof simple
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