arXiv subjects

Hiroshi Fukuda

Publications and source records attributed to Hiroshi Fukuda.

18 recordsLinked to original sources

arXiv2026

Fold of a bifurcation solution from the figure-eight choreography in the three body problem

In the figure-eight choreography in the classical three-body problem, both-side bifurcation solutions sometimes fold on one side of the bifurcation point with cusp of action. Three numerical examples of a such fold for figure-eight choreography under the Lennard-Jones-type potential and one under the homogeneous potential are introduced. Up to the fourth order of representation variable of the Lyapunov-Schmidt reduced action in two dimensions with three-fold symmetry, the fold is analyzed. It is shown that the bifurcation solutions fold under a condition given by the third and fourth expansion coefficients.

math-ph↗

arXiv2025

Bifurcation analysis of figure-eight choreography in the three-body problem based on crystallographic point groups

The bifurcation of figure-eight choreography is analyzed by its symmetry group based on the variational principle of the action. The irreducible representations determine the symmetry and the dimension of the Lyapunov-Schmidt reduced action, which yields four types of bifurcations in the sequence of the bifurcation cascade. Type 1 bifurcation, represented by trivial representation, bifurcates two solutions. Type 2, by non-trivial one-dimensional representation, bifurcates two congruent solutions. Type 3 and 4, by two-dimensional irreducible representations, bifurcate two sets of three and six congruent solutions, respectively. We analyze numerical bifurcation solutions previously published and four new ones: non-symmetric choreographic solution of type 2, non-planar solution of type 2, $y$-axis symmetric solution of type 3, and non-symmetric solution of type 4.

math-ph↗

arXiv2020

Variational principle of action and group theory for bifurcation of figure-eight solutions

Figure-eight solutions are solutions to planar equal mass three-body problem under homogeneous or inhomogeneous potentials. They are known to be invariant under the transformation group $D_6$: the dihedral group of regular hexagons. Numerical investigation shows that each figure-eight solution has some bifurcation points. Six bifurcation patterns are known with respect to the symmetry of the bifurcated solution. In this paper we will show the followings. The variational principle of action and group theory show that the bifurcations of every figure-eight solution are determined by the irreducible representations of $D_6$. Each irreducible representation has one to one correspondence to each bifurcation. This explains numerically observed six bifurcation patterns. In general, in Lagrangian mechanics, bifurcations of a periodic solution is determined by irreducible representations of the transformation group that leaves this solution invariant.

math-ph↗

arXiv2019

Variational principle for bifurcation in Lagrangian mechanics

An application of variational principle to bifurcation of periodic solution in Lagrangian mechanics is shown. A few higher derivatives of the action integral at a periodic solution reveals the behaviour of the action in function space near the solution. Then the variational principle gives a method to find bifurcations from the solution. The second derivative (Hessian) of the action has an important role. At a bifurcation point, an eigenvalue of Hessian tends to zero. Inversely, if an eigenvalue tends to zero, the zero point is a bifurcation point. The third and higher derivatives of the action determine the properties of the bifurcation and bifurcated solution.

physics.class-ph↗

arXiv2019

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

We report on the Morse index and periodic solutions bifurcating from the figure-eight choreography for the equal mass three-body problem under homogeneous potential $-1/r^a$ for $a \ge 0$, and under Lennard-Jones (LJ) type potential $1/r^{12}-1/r^6$, where $r$ is a distance between bodies. It is shown that the Morse index changes at a bifurcation point and all solutions bifurcating are approximated by variational functions responsible for the change of the Morse index. Inversely we observed %numerically bifurcation occurs at every point where the Morse index changes for the figure-eight choreography under $-1/r^a$, and for $α$ solution under LJ type potential, where $α$ solution is a figure-eight choreography tending to that under $-1/r^6$ for infinitely large period. Thus, to our numerical studies, change of the Morse index is not only necessary but also sufficient condition for bifurcation for these choreographies. Further we observed that the change of the Morse index is equal to the number of bifurcated solutions regarding solutions with congruent orbits as the same solution.

math-ph↗

arXiv2018

Decomposition of the Hessian matrix for action at choreographic three-body solutions with figure-eight symmetry

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 solution is a periodic solution to equal mass planar three-body problem under potential function $\sum_{i<j} U(r_{ij})$, in which three masses chase each other on a single closed loop with equal time delay. We treat choreographic solutions that have the same symmetries as the figure-eight, namely, symmetry for choreography, for time reversal, and for time shift of half period. The function space of periodic functions are decomposed into five subspaces by these symmetries. Namely, one subspace of trivial oscillators with eigenvalue $4π^2/T^2\times k^2$, $k=0,1,2,\dots$, four subspaces of choreographic functions, and four subspaces of "zero-choreographic" functions. Therefore, the matrix representation of the Hessian is also decomposed into nine corresponding blocks. Explicit expressions of base functions and the matrix representation of the Hessian for each subspaces are given. The trivial eigenvalues with $k\ne 0$ are quadruply degenerated, while with $k=0$ are doubly degenerated that correspond to the conservation of linear momentum in $x$ and $y$ direction. The eigenvalues in choreographic subspace have no degeneracy in general. In "zero-choreographic" subspace, every eigenvalues are doubly degenerated.

math-ph↗

arXiv2018

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

We report on numerical calculations of Morse index for figure-eight choreographic solutions to a system of three identical bodies in a plane interacting through homogeneous potential, $-1/r^a$, or through Lennard-Jones-type (LJ) potential, $1/r^{12} - 1/r^6$, where $r$ is a distance between the bodies. The Morse index is a number of independent variational functions giving negative second variation $S^{(2)}$ of action functional $S$. We calculated three kinds of Morse indices, $N$, $N_c$ and $N_e$, in the domain of the periodic, the choreographic and the figure-eight choreographic function, respectively. For homogeneous system, we obtain $N=4$ for $0 \le a < a_0$, $N=2$ for $a_0 < a < a_1$, $N=0$ for $a_1 < a$, and $N_c=N_e=0$ for $0 \le a$, where $a_0=0.9970$ and $a_1=1.3424$. For $a=1$, we show a strong relationship between the figure-eight choreography and the periodic solution found by Simó through the $S^{(2)}$. For LJ system, we calculated the index for the solution tending to the figure-eight solution of $a=6$ homogeneous system for the period $T \to \infty$. We obtain $N$, $N_c$ and $N_e$ as monotonically increasing functions of the gradual change in $T$ from $T \to \infty$, which start with $N=N_c=N_e=0$, jump at the smallest $T$ by $1$, and reach $N=12$, $N_c=4$, and $N_e=1$ for $T \to \infty$ in the other branch.

math-ph↗

arXiv2017

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

We report on figure-eight choreographic solutions to a system of three identical particles interacting through a potential of Lennard-Jones type, $1/r^{12}-1/r^6$ where $r$ is a distance between the particles. By numerical search, we found there are a multitude of such solutions. A series of them are close to a figure-eight solutions to a homogeneous system with no $1/r^{12}$ term in the potential. The rest are very different from them and have several points with large curvatures in their figure-eight orbits, at which particles are repelled. Here figure-eight choreographies are the periodic motion whose shape is symmetric in both horizontal and vertical axis, starting with an isosceles triangle configuration and going back to an isosceles triangle configuration with opposite direction through Euler configuration. Thus the lobe of this figure-eight may be complex shape and needs not to be convex.

math.DS↗

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

A monolithically integrated polarization entangled photon pair source on a silicon chip

Integrated photonic circuits are one of the most promising platforms for large-scale photonic quantum information systems due to their small physical size and stable interferometers with near-perfect lateral-mode overlaps. Since many quantum information protocols are based on qubits defined by the polarization of photons, we must develop integrated building blocks to generate, manipulate, and measure the polarization-encoded quantum state on a chip. The generation unit is particularly important. Here we show the first integrated polarization-entangled photon pair source on a chip. We have implemented the source as a simple and stable silicon-on-insulator photonic circuit that generates an entangled state with 91 \pm 2% fidelity. The source is equipped with versatile interfaces for silica-on-silicon or other types of waveguide platforms that accommodate the polarization manipulation and projection devices as well as pump light sources. Therefore, we are ready for the full-scale implementation of photonic quantum information systems on a chip.

quant-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↗

arXiv2011

Polyominoes and Polyiamonds as Fundamental Domains of Isohedral Tilings with Rotational Symmetry

We describe computer algorithms that produce the complete set of isohedral tilings by n-omino or n-iamond tiles in which the tiles are fundamental domains and the tilings have 3-, 4-, or 6-fold rotational symmetry. The symmetry groups of such tilings are of types p3, p31m, p4, p4g, and p6. There are no isohedral tilings with symmetry groups p3m1, p4m, or p6m that have polyominoes or polyiamonds as fundamental domains. We display the algorithms' output and give enumeration tables for small values of n. This expands on our earlier works (Fukuda et al 2006, 2008).

cs.CG↗

arXiv2009

Three-Body Choreographies in Given Curves

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 cases, the shape of the orbit, conservation of the centre of mass and a constant of motion (the angular momentum or the total energy) determines the motion of the three bodies. We show, by a geometrical method, that choreographic motions, in which equal mass three bodies chase each other around a same curve, will be uniquely determined for the following two cases. (i) Convex curves that have point symmetry and non-vanishing angular momentum are given. (ii) Eight-shaped curves which are similar to the curve for the figure-eight solution and the energy constant are given. The reality of the motion should be tested whether the motion satisfies an equation of motion or not. Extensions of the method for generic curves are shown. The extended methods are applicable to generic curves which does not have point symmetry. Each body may have its own curve and its own non-vanishing masses.

math-ph↗

arXiv2007

Entanglement generation using silicon wire waveguide

We report the first entanglement generation experiment that utilizes a silicon waveguide. Using spontaneous four-wave mixing in a 1.09-cm-long silicon wire waveguide, we generated 1.5-um, high-purity time-bin entangled photons without temperature control, and observed a two-photon interference fringe with >73% visibility.

quant-ph↗

arXiv2004

Synchronised Similar Triangles for Three-Body Orbit with Zero Angular Momentum

Geometrical properties of three-body orbits with zero angular momentum are investigated. 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 (``synchronised similar triangles''). This similarity yields kinematic equalities between mutual distances and magnitude of momenta. Moreover, if the orbit is a solution to the equation of motion under homogeneous potential, the orbit has a new constant involving momenta. For orbits with zero angular momentum and non-constant moment of inertia, we introduce scaled variables, positions divided by square root of the moment of inertia and momenta derived from the velocity of the scaled positions. Then the similarity and the kinematic equalities hold for the scaled variables. Using this similarity, we prove that any bounded three-body orbit with zero angular momentum under homogeneous potential whose degree is smaller than 2 has infinitely many collinear configurations (syzygies or eclipses) or collisions.

math-ph↗

arXiv2003

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 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 is 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 alpha \neq -2, 2, 4 is impossible. And motions which satisfy the same conditions for alpha=2, 4 are solved explicitly. Shapes of these orbits are not figure-eight and these motions have collision. Therefore non-conservation of the moment of inertia for figure-eight choreography for alpha \neq -2 is proved. We also prove that the motion which satisfies conditions (I)-(III) with general masses under the Newtonian potential alpha=-1 is impossible.

math-ph↗

arXiv2002

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.

math-ph↗