Saari's homographic conjecture claims that, in the N-body problem under the
homogeneous potential, for , a motion having constant configurational measure is
homographic, wh...

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 potentia...

2012 J. Phys. A: Math. Theor. 45 045208 Jan 2012 [Refereed]

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 pote...

As shown by Johannes Kepler in 1609, in the two-body problem, the shape of
the orbit, a given ellipse, and a given non-vanishing constant angular momentum
determines the motion of the planet completely.
Even in the three-body problem, in some ca...

Saari's homographic conjecture, which extends a classical statement proposed
by Donald Saari in 1970, claims that solutions of the Newtonian -body
problem with constant configurational measure are homographic. In other words,
if the mutual dist...

Pacific Journal of Mathematics 219(2) 271-283 2005 [Refereed]

The figure eight is a remarkable solution to the Newtonian three-body problem in which the three equal masses chase each around a planar curve having the qualitative shape and symmetries of a figure eight. Here we prove that each lobe of this eigh...

Journal of Physics A: Mathemathical and General 37(44) 10571-10584 Oct 2004 [Refereed]

If the moment of inertia is also constant along the orbit, the triangle whose vertexes are the positions of the bodies, and the triangle whose perimeters are the momenta of the bodies, are always similar ('synchronized similar triangles'). This si...

Nuclear Physics B - Proceedings Supplements 129--130 774-776 Mar 2004

The new algorithm of the finite lattice method is applied to generate the high-temperature expansion series of the simple cubic Ising model to β^{50} for the free energy, to β^{32} for the magnetic susceptibility and to β^{29} for the second momen...

Journal of Physics A: Mathematical and General, volume 36, issue 42, pages 10537 - 10549, (2003) 36 Oct 2003 [Refereed]

We investigate three-body motion in three dimensions under the interaction
potential proportional to r^alpha (alpha \neq 0) or log r, where r represents
the mutual distance between bodies, with the following conditions: (I) the
moment of inertia i...

We propose a new algorithm of the finite lattice method to generate the
high-temperature series for the Ising model in three dimensions. It enables us
to extend the series for the free energy of the simple cubic lattice from the
previous series of...

J. Phys. A36, 2791--2800 (2003) 36(11) 2791-2800 Mar 2003 [Refereed]

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...

We developed a method to calculate the eigenvalues and eigenfunctions of the second derivative (Hessian) of action at choreographic three-body solutions that have the same symmetries as the figure-eight solution. A choreographic three-body solutio...

We show that the lemniscate curve is the simplest candidate of the orbit for the body figure eight choreography. Here figure eight choreography is the periodic motion of equal masses with equal time spacing o...