Toshiaki Fujiwara

Algorithm of the finite-lattice method for high-temperature expansion of the Ising model in three dimensions

We propose an 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 26th order to 46th order in the inverse temperature. The obtained series give the estimate of the critical exponent for the specific heat in high precision.

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) = ( + cn(t))sn(t)/(1 + cn2(t)) parametrized by the Jacobian elliptic functions sn and cn with modulus k2 = (2 + √3)/4, conserve the centre of mass and the angular momentum, where and are the orthogonal unit vectors defining the plane of the motion. They also conserve the moment of inertia, the kinetic energy, the sum of squares of the curvature, the product of distances and the sum of squares of distances between bodies. We find that they satisfy the equation of motion under the potential energy ∑i<j((1/2) ln rij − (√3/24)rij2) or ∑i<j(1/2) ln rij − ∑i(√3/8)ri2, where rij is the distance between bodies i and j, and ri the distance from the origin. The first term of the potential energies is the universal gravitation in two dimensions but the second term is a 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.

Saari’s homographic conjecture of the three-body problem

Saaris homographic conjecture, which extends a classical statement proposed by Donald Saari in 1970, claims that solutions of the Newtonian n-body problem with constant configurational measure are homographic. In other words, if the mutual distances satisfy a certain relationship, the configuration of the particle system may change size and position but not shape. We prove this conjecture for large sets of initial conditions in three-body problems given by homogeneous potentials, including the Newtonian one. Some of our results are true for n > 3.

Evolution of the moment of inertia of three-body figure-eight choreography

We investigate three-body motion in three dimensions under the interaction potential proportional to rα (α ≠ 0) or log r, where r represents the mutual distance between bodies, with the following conditions: (I) the moment of inertia is a non-zero constant, (II) the angular momentum is zero and (III) one body is on the centre of mass at an instant. We prove that the motion which satisfies conditions (I)–(III) with equal masses for α ≠ −2, 2, 4 is impossible. And motions which satisfy the same conditions for α = 2, 4 are solved explicitly. Shapes of these orbits are not figure-eight and these motions have collision. Therefore, the moment of inertia for figure-eight choreography for α ≠ −2 is proved to be inconstant along the orbit. We also prove that the motion which satisfies conditions (I)–(III) with general masses under the Newtonian potential α = −1 is impossible.

Saari’s homographic conjecture for a planar equal-mass three-body problem under the Newton gravity

Saari?s homographic conjecture in an N-body problem under the Newton gravity is as follows. Configurational measure , which is the product of the square root of the moment of inertia I = (?mk)?1?mimjr2ij and the potential function U = ?mimj/rij, is constant if and only if the motion is homographic. Where mk represents the mass of the body k and rij represents the distance between bodies i and j. We prove this conjecture for the planar equal-mass three-body problem. In this work, we use three sets of shape variables. In the first step, we use ? = 3q3/(2(q2 ? q1)), where represents the position of the body k. Using r1 = r23/r12 and r2 = r31/r12 in the intermediate step, we finally use ? itself and ? = I3/2/(r12r23r31). The shape variables ? and ? make our proof simple.

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

Saari conjectured that the N-body motion with a constant configurational measure is a motion with fixed shape. Here, the configurational measure ? is a scale-invariant product of the moment of inertia I = ?kmk|qk|2 and the potential function U = ?i 0. Namely, ? = I?/2U. We will show that this conjecture is true for a planar equal-mass three-body problem under the strong force potential ?i < j1/|qi ? qj|2.

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

Saaris homographic conjecture claims that, in the N-body problem under the homogeneous potential,

Morse index for figure-eight choreographies of the planar equal mass three-body problem

U=\alpha^{-1}\sum m_i m_j/r_{ij}^\alpha

Figure-eight choreographies of the equal mass three-body problem with Lennard-Jones-type potentials

for

Determination of motion from orbit in the three-body problem

\alpha\ne 0