arXiv subjects

Tetsuya Taniguchi

Publications and source records attributed to Tetsuya Taniguchi.

3 recordsLinked to original sources

arXiv2015

Saari's homographic conjecture for general masses in planar three-body problem under Newton potential and a strong force potential

Saari's homographic conjecture claims that, in the N-body problem under the homogeneous potential, $U=α^{-1}\sum m_i m_j/r_{ij}^α$ for $α\ne 0$, a motion having constant configurational measure $μ=I^{α/2}U$ is homographic, where $I$ represents the moment of inertia defined by $I=\sum m_i m_j r_{ij}^2/\sum m_k$, $m_i$ the mass, and $r_{ij}$ the distance between particles. We prove this conjecture for general masses $m_k>0$ in the planar three-body problem under Newton potential ($α=1$) and a strong force potential ($α=2$).

math-ph↗

arXiv2012

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 μ=\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 ζ=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 μitself and ρ=I^{3/2}/(r_{12}r_{23}r_{31}). The shape variables μand ρmake our proof simple.

math-ph↗

arXiv2011

Saari's homographic conjecture for planar equal-mass three-body problem under a strong force potential

Donald Saari conjectured that the $N$-body motion with constant configurational measure is a motion with fixed shape. Here, the configurational measure $μ$ is a scale invariant product of the moment of inertia $I=\sum_k m_k |q_k|^2$ and the potential function $U=\sum_{i 0$. Namely, $μ= I^{α/2}U$. We will show that this conjecture is true for planar equal-mass three-body problem under the strong force potential $\sum_{i<j} 1/|q_i-q_j|^2$.

math-ph↗